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In mathematics, the empty set or void set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic...

mathematical field of point-set topology, a continuum (plural: "continua") is a nonempty compact connected metric space, or, less frequently, a compact connected...

{\displaystyle I} -indexed family ( S i ) i ∈ I {\displaystyle (S_{i})_{i\in I}} of nonempty sets, there exists an I {\displaystyle I} -indexed set ( x i ) i ∈ I {\displaystyle...

every d + 1 {\displaystyle d+1} of these sets is nonempty, then the whole collection has a nonempty intersection; that is, ⋂ j = 1 n X j ≠ ∅ . {\displaystyle...

efforts to extend Schauder's work. Schauder fixed-point theorem: Let C be a nonempty closed convex subset of a Banach space V. If f : C → C is continuous with...

space is a topological space X that contains no nonempty dense-in-itself subset. Equivalently, every nonempty subset A of X contains a point isolated in A...

{\displaystyle (\star )} on a nonempty set H {\displaystyle H} is a mapping from H × H {\displaystyle H\times H} to the nonempty power set P ∗ ( H ) {\displaystyle...

R\cup S} ∪ {\displaystyle \cup }  of increasing nonempty chain ∪ {\displaystyle \cup }  of increasing nonempty chain Arbitrary unions (of at least 1 set) Arbitrary unions...

set of a bracket-generating system has nonempty interior or, equivalently, that any attainable set has nonempty interior in the topology of the corresponding...

implying P {\displaystyle P} is nonempty. However, in practice, one often checks (1) and then verifies (2) only for nonempty chains, since the case of the...

{\displaystyle F\colon X\to Y} be a lower hemicontinuous set-valued function with nonempty convex closed values. Then there exists a continuous selection f : X →...

{\displaystyle \operatorname {Ran} (X)} whose underlying set is the set of all nonempty finite subsets of X: for a metric space X the topology is induced by the...

In mathematics, the simplex category (or simplicial category or nonempty finite ordinal category) is the category of non-empty finite ordinals and order-preserving...

of balls (in the sense of the metric induced by the absolute value) is nonempty: B 1 ⊇ B 2 ⊇ ⋯ ⇒ ⋂ n ∈ N B n ≠ ∅ . {\displaystyle B_{1}\supseteq B_{2}\supseteq...

{\displaystyle i\in I} . The axiom of choice implies that any product of nonempty sets is nonempty. Given two sets ⁠ E {\displaystyle E} ⁠ and ⁠ F {\displaystyle...

numbers. X is nonempty and for every ω-sequence of elements of X, there exists at least one element of X not included in it. That is, X is nonempty and there...

space X such that every (nonempty) irreducible closed subset of X is the closure of exactly one point of X: that is, every nonempty irreducible closed subset...

By a theorem of L. E. J. Brouwer, this is equivalent to being perfect, nonempty, compact, metrizable and zero-dimensional. The Cantor ternary set C {\displaystyle...

Cantor set. Every decreasing nested sequence of nonempty closed subsets S1 ⊇ S2 ⊇ ... in (X, d) has a nonempty intersection. Every increasing nested sequence...

meaning that every descending nested sequence of internal sets has a nonempty intersection. Let κ be a finite or infinite cardinal number and M a model...

In mathematics, a nonempty subset S of a group G is said to be symmetric if it contains the inverses of all of its elements. In set notation a subset S...

conditionally complete, i.e., which "lacks gaps" in the sense that every nonempty subset with an upper bound has a least upper bound in the set. More symbolically:...

-dimensional centrally symmetric polytope have at least 3 d {\displaystyle 3^{d}} nonempty faces? More unsolved problems in mathematics In geometry, more specifically...

numbers. A Dedekind cut is a partition of the rational numbers into two nonempty sets A and B, such that each element of A is less than every element of...

complicated. One family of basis theorems concern nonempty effectively closed sets (that is, nonempty Π 1 0 {\displaystyle \Pi _{1}^{0}} sets in the arithmetical...

of the edge sets of the given graphs. Any disjoint union of two or more nonempty graphs is necessarily disconnected. The disjoint union is also called the...

mathematics, an algebraic structure or algebraic system consists of a nonempty set A (called the underlying set, carrier set or domain), a collection...

tournament solution is a function that maps an oriented complete graph to a nonempty subset of its vertices. It can informally be thought of as a way to find...

the associative and identity element properties are optional. In fact, a nonempty associative quasigroup is a group. A quasigroup that has an identity element...

a partially ordered set that has a join (a least upper bound) for any nonempty finite subset. Dually, a meet-semilattice (or lower semilattice) is a partially...

parts of the vocabulary under a formal grammar. Vocabulary is a finite, nonempty set of symbols. Terminal symbols are symbols that cannot be replaced by...

has a nonempty perfect subset (Kechris 1995, p. 150). Note that having the perfect set property is not the same as being a perfect set. As nonempty perfect...

states that for any continuous function f {\displaystyle f} mapping a nonempty compact convex set to itself, there is a point x 0 {\displaystyle x_{0}}...

set of nonempty sets has a choice function. A weaker form of AC, the axiom of countable choice (ACω) states that every countable set of nonempty sets has...

numbers. The axiom of choice is equivalent to the statement that every nonempty set of cardinal numbers has a minimum element (which is also the infimum...

modern logic only the contradictories in the square of opposition apply, because domains may be empty. (Black areas are empty, red areas are nonempty.)...

the axiom of choice, the following statements are equivalent: For every nonempty set X there exists a binary operation • such that (X, •) is a group. The...

uniformly convex Banach spaces. It asserts that if K {\displaystyle K} is a nonempty convex closed bounded set in uniformly convex Banach space and f {\displaystyle...

formulation of AC is that the Cartesian product of a family of nonempty sets is nonempty; but since the empty set is most certainly compact, the proof...

also shows that every nonempty complete metric space with no isolated point is uncountable. (If X {\displaystyle X} is a nonempty countable metric space...

{\displaystyle E} is a normed vector space and K {\displaystyle K} is a nonempty convex subset of E {\displaystyle E} that is compact under the weak topology...

region with two vertical sides and one horizontal side (if the region is nonempty) can be found as the nearest common ancestor of the leftmost and rightmost...

Discrete Mathematics. PWS-Kent. p. 114. ISBN 0-53492-373-9. An alphabet is a nonempty finite set the members of which are called symbols or characters. Ebbinghaus...

In mathematics, a nonempty collection of sets is called a 𝜎-ring (pronounced sigma-ring) if it is closed under countable union and relative complementation...

an isomorphism from a nonempty open subset of X to a nonempty open subset of Y, and vice versa: an isomorphism between nonempty open subsets of X, Y by...

{\displaystyle \mathbb {R} } that splits every nonempty open set into two nonmeagre sets. That is, for every nonempty open set U ⊆ R {\displaystyle U\subseteq...

notion is the intersection of an arbitrary nonempty collection of sets. If M {\displaystyle M} is a nonempty set whose elements are themselves sets, then...

formula of monadic predicate calculus is logically valid (true for all nonempty domains). Adding a single binary relation symbol to monadic logic, however...

In mathematics, a semigroup is a nonempty set together with an associative binary operation. A special class of semigroups is a class of semigroups satisfying...

infinite set is infinite. The Cartesian product of an infinite set and a nonempty set is infinite. The Cartesian product of an infinite number of sets, each...

equivalent: No two nonempty open sets are disjoint. X cannot be written as the union of two proper closed subsets. Every nonempty open set is dense in...

directed both upward and downward. Directed sets are a generalization of nonempty totally ordered sets. That is, all totally ordered sets are directed sets...

in the areas of combinatorics and computer science, a Lyndon word is a nonempty string that is strictly smaller in lexicographic order than all of its...

{\displaystyle V_{i}\to X,i=1,2} morphisms of varieties, G contains a nonempty open subset such that for each g in the set, either g V 1 × X V 2 {\displaystyle...

theorem—Let A {\displaystyle A} and B {\displaystyle B} be two disjoint nonempty convex subsets of R n {\displaystyle \mathbb {R} ^{n}} . Then there exist...

mathematics, a topological space is said to be ultraconnected if no two nonempty closed sets are disjoint. Equivalently, a space is ultraconnected if and...

ordinal is the well-ordered set of all smaller ordinals. The union of a nonempty set of ordinals that has no greatest element is then always a limit ordinal...

be of infinite dimension. It asserts that if K {\displaystyle K} is a nonempty convex closed subset of a Hausdorff locally convex topological vector space...

In the mathematical field of graph theory, the bondage number of a nonempty graph G is the cardinality of the smallest set of edges whose removal results...

product is nonempty. That the product is nonempty follows immediately from the axiom of choice and the fact that the factors are nonempty. For each i...

complement is δ-closed. Dense set A set is dense if it has nonempty intersection with every nonempty open set. Equivalently, a set is dense if its closure...

Complex functions are generally assumed to have a domain that contains a nonempty open subset of the complex plane. For any complex function, the values...

=0\},} where A 1 , … , A n {\displaystyle A_{1},\ldots ,A_{n}} are finite nonempty subsets of a field F and P ( x 1 , … , x n ) {\displaystyle P(x_{1},\ldots...

{\displaystyle (x_{n})_{n=1}^{\infty }} of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence...

data structures (structure in the sense of mathematical logic, that is a nonempty set together with a number of functions (operations) and relations over...

partially ordered set that satisfies the hypothesis of Zorn's lemma when nonempty. In descriptive set theory, an inductive set of real numbers (or more generally...

(+) 1; count := count + 1; nonempty.signal end append; procedure remove(result x: portion) ; begin if count = 0 then nonempty.wait; note 0 < count <= N;...

\emptyset }} is the category with nonempty sets as objects and functions as morphisms. This implies not only that every nonempty rectangular band is isomorphic...

*A_{n})} Countable saturation: If {Ak}k ∈ N is a decreasing sequence of nonempty internal sets, with k ranging over the natural numbers, then ⋂ k A k ≠...

complete lattice satisfies at least one of these properties for bounded and nonempty subsets. For comparison, in a general lattice, only pairs of elements need...

branches of mathematics, Cantor's isomorphism theorem states that every two nonempty countable dense unbounded linear orders are order-isomorphic. The theorem...

also be called null set. In measure theory, a null set is a (possibly nonempty) set with zero measure. A null space of a mapping is the part of the domain...

(}f^{(k)}-g^{(k)}{\bigr )}(z)=0\right\}.} We'll show T {\displaystyle T} is nonempty, open, and closed. Then by connectedness of D {\displaystyle D} , T {\displaystyle...

(X;A,B)} be an excisive triad with C = A ∩ B {\displaystyle C=A\cap B} nonempty, and suppose the pair ( A , C ) {\displaystyle (A,C)} is ( m − 1 {\displaystyle...

must also be semiregular. X is quasi-regular if for any nonempty open set G, there is a nonempty open set H such that the closure of H is contained in G...

the smallest size of a hitting set, a set of vertices of G that has a nonempty intersection with each of the subgraphs. Brambles may be used to characterize...

theory. The problem asks if a list of deterministic finite automata has nonempty intersection. A non-emptiness decision problem is defined (for a particular...

} ¯ {\displaystyle \bigcap _{n}{\overline {\{x_{m}\mid m\geq n\}}}} is nonempty. Indeed, a limit of a convergent subsequence is necessarily in the above...

Banach space; F is lower hemicontinuous; for all x in X, the set F(x) is nonempty, convex and closed. The approximate selection theorem states the following:...

after John Tukey and Oswald Teichmüller, is a lemma that states that every nonempty collection of finite character has a maximal element with respect to inclusion...

has a unique nonempty compact (closed and bounded) fixed set S. One way of constructing a fixed set is to start with an initial nonempty closed and bounded...

\ldots ,\psi _{m}} , the open set condition requires that there exists a nonempty, open set V satisfying two conditions: ⋃ i = 1 m ψ i ( V ) ⊆ V , {\displaystyle...

array is empty, since an empty array has no maximum nonempty subarray. If the array is nonempty, its first element could be used in place of negative...

all intersections of all but one polygon are nonempty, then the intersection of all the polygons is nonempty. Krein–Milman theorem: A convex polygon is...

extends outward in all the directions given by the recession cone. Given a nonempty set A ⊂ X {\displaystyle A\subset X} for some vector space X {\displaystyle...

\mathbb {X} } be a complete metric space. Suppose L {\displaystyle L} is a nonempty, compact subset of X {\displaystyle \mathbb {X} } and let ε > 0 {\displaystyle...

∣ ξ < α } {\displaystyle A\setminus \{a_{\xi }\mid \xi <\alpha \}} is nonempty, or leaves a α {\displaystyle a_{\alpha }} undefined if the complement...

{\displaystyle 1} . Now assume for a contradiction that Z {\displaystyle Z} is nonempty. Then it has a least upper bound c {\displaystyle c} , which is also positive...

In mathematics, a filter on a set is a collection of nonempty subsets which is upward closed and closed under finite intersections (see Filter on a set...

space X is said to be disconnected if it is the union of two disjoint nonempty open sets. Otherwise, X is said to be connected. A subset of a topological...

its elements. A general property of finite groups implies that a finite nonempty subset of a symmetric group is a permutation group if and only if it is...

and an index im such that Sm has m elements and such that every sum of a nonempty subset of Sm belongs to Nim. Schur's theorem in Ramsey theory states that...

{\displaystyle \beta } may be empty, but γ {\displaystyle \gamma } must be nonempty. The rule S → ϵ {\displaystyle S\rightarrow \epsilon } is allowed if S...

begins the game by picking a nonempty open subset I 0 ⊆ Y {\displaystyle I_{0}\subseteq Y} , and player II responds with a nonempty open subset J 0 ⊆ I 0 {\displaystyle...

a set of n {\displaystyle n} labelled objects into k {\displaystyle k} nonempty unlabelled subsets. Equivalently, they count the number of different equivalence...

X} is a Baire space if and only if Player I has no winning strategy. A nonempty topological space X {\displaystyle X} in which Player II has a winning...

and let F be a filter on P; that is, F is a subset of P such that: F is nonempty If p, q ∈ P and p ≤ q and p is an element of F, then q is an element of...

it is countable. Every nonempty, paracompact, connected 1-manifold is homeomorphic either to R or the circle. Every nonempty, compact, connected 2-manifold...

defined without recourse to model theory. Here we define a structure on a nonempty set M {\displaystyle M} in a set-theoretic manner, as a sequence ( S n...

others losing coalitions. It is sometimes assumed that a simple game is nonempty or that it does not contain an empty set. However, in other areas of mathematics...

y\in P,D(y)].\,} where D is an arbitrary predicate and P is an arbitrary nonempty set. The proof begins by recognizing it is true that either everyone in...

the fact that the spectrum of any element of a complex Banach algebra is nonempty: for every element a {\displaystyle a} of a complex Banach algebra A {\displaystyle...

element x 0 {\displaystyle x_{0}} in X (which exists as X is supposed to be nonempty), and one defines g by g ( y ) = x {\displaystyle g(y)=x} if y = f ( x...

abstract setting, it is possible to have subspaces without points but still nonempty. The parts of the paradoxical decomposition do intersect in the sense of...

says that in every KKM covering, the common intersection of all n sets is nonempty, i.e.: ⋂ i = 1 n C i ≠ ∅ . {\displaystyle \bigcap _{i=1}^{n}C_{i}\neq \emptyset...

any two objects A and B, there is only one morphism from A to B. Any two nonempty indiscrete categories are equivalent to each other. The functor from Set...

characterization can be formulated for prime ideals in noncommutative rings. A nonempty subset S ⊆ R is called an m-system if for any a and b in S, there exists...

intersection, but removing any set from this subfamily causes it to have a nonempty intersection. Therefore, it is a minimal subfamily with an empty intersection...

the naturals are well-ordered—every nonempty subset of N has a least element—one can reason as follows. Let a nonempty X ⊆ N be given and assume X has no...

this topology is not Hausdorff (T2), regular or normal because no two nonempty open sets are disjoint (that is, it is hyperconnected). The double-pointed...

realization of Ki, and denote the nerve of {U1, ... , Un } by N. If, for each nonempty J ⊂ I {\displaystyle J\subset I} , the intersection ⋂ i ∈ J U i {\displaystyle...

the general notion. A concrete Boolean algebra or field of sets is any nonempty set of subsets of a given set X closed under the set operations of union...

D ∖ { c } ) ∖ c {\displaystyle \cap (D\setminus \{c\})\setminus c} is nonempty. This concept was originally proposed by M. Gereb-Graus in "Complexity...

u} such that t = s u {\displaystyle t=su} . If u {\displaystyle u} is nonempty, s {\displaystyle s} is said to be a proper prefix of t {\displaystyle...

has the property that all pairwise intersections are nonempty, then the whole family has a nonempty intersection. Bandelt & Chepoi (2008), 1. Basic notions...

ordered field is a partition of it, (A, B), such that A is nonempty and closed downwards, B is nonempty and closed upwards, and A contains no greatest element...

aforementioned fact that the rational numbers are not locally connected. A nonempty product space ∏ i X i {\displaystyle \prod _{i}X_{i}} is locally connected...

unzerlegbar) is the principle that the continuum cannot be partitioned into two nonempty pieces. This principle was established by Brouwer in 1928 using intuitionistic...

a topological space is feebly compact if every locally finite cover by nonempty open sets is finite. The concept was introduced by Sibe Mardešić and P...

{C}}_{\varepsilon }(X)} if the set of ε-balls centered at points of σ has a nonempty intersection. In other words, the Čech complex is the nerve of the set...

and ax + by > 0}. The set S is nonempty since it contains either a or –a (with x = ±1 and y = 0). Since S is a nonempty set of positive integers, it has...

upward-directed family G ⊆ Σ {\displaystyle {\mathcal {G}}\subseteq \Sigma } of nonempty open sets such that its union is in Σ , {\displaystyle \Sigma ,} the measure...

nonempty and convex, then its relative interior r e l i n t ( A ) {\displaystyle \mathrm {relint} (A)} is the union of a nested sequence of nonempty compact...

(Predicate term). In the Venn diagrams, black areas are empty and red areas are nonempty. White areas may or may not be empty. The faded arrows and faded red areas...

n} . A partition of a set S {\displaystyle S} is defined as a family of nonempty, pairwise disjoint subsets of S {\displaystyle S} whose union is S {\displaystyle...

is the set's cardinality, denoted as | S | {\displaystyle |S|} . If a nonempty set is finite, its elements may be written in a sequence: x 1 , x 2 , …...

lower set that is closed under finite joins (suprema); that is, it is nonempty and for all x, y in I, the element x ∨ y {\displaystyle x\vee y} of P is...

Iwasawa. In voting theory, he is known for the Smith set, the smallest nonempty set of candidates such that, in every pairwise matchup (two-candidate election/runoff)...

side and the right-hand side are equal. Next, consider any nonempty list I. Since I is nonempty, it has a head item, x, and a tail list, xs, so we can express...

c) ≡ (a ∧ b) ∨ (¬a ∧ c) The power set (set of all subsets) of any given nonempty set S forms a Boolean algebra, an algebra of sets, with the two operations...

Thomas Schwartz) is a result about voting systems designed to choose a nonempty set of winners from the preferences of certain individuals, where each...

model above). The languages are treated as distinct sets, but as sets with nonempty intersections. More fundamentally, The Intermediate also rejects linguistic...

of sets, both referring to certain families of sets. In order theory, a nonempty family of sets R {\displaystyle {\mathcal {R}}} is called a ring (of sets)...

{\displaystyle X} be a Banach space and A {\displaystyle A} a weakly closed nonempty subset of X . {\displaystyle X.} The following conditions are equivalent:...

a topological space X is a family A {\displaystyle {\mathfrak {A}}} of nonempty open sets Uα such that not two of them are equal and they are that their...

algebraic constructions that yield vector spaces related to given ones. A nonempty subset W {\displaystyle W} of a vector space V {\displaystyle V} that is...

\operatorname {Lin} (\mathbb {R} ^{n};\mathbb {R} ^{d})} is a map from some open nonempty set U ⊂ R d {\displaystyle U\subset \mathbb {R} ^{d}} , where Lin ⁡ ( R...

a nonempty subset of X. If x is a point of X, the distance of x from A is defined as d(x, A) = inf{ d(x, a): a in A}. If A and B are both nonempty subsets...

vector x {\displaystyle x} in a Hilbert space H {\displaystyle H} and every nonempty closed convex C ⊆ H , {\displaystyle C\subseteq H,} there exists a unique...

always true. This is related to the concept of a vacuous truth. Every nonempty (disjunctive) clause is logically equivalent to an implication of a head...

indexed by an element of U {\displaystyle U} , All intersections of all nonempty families of elements of U {\displaystyle U} indexed by an element of U...

axioms. An affine plane geometry is a nonempty set X (whose elements are called "points"), along with a nonempty collection L of subsets of X (whose elements...

consequence, we can modify the axiom of extensionality to apply only to nonempty sets, so that it reads: ∀ A ∀ B ( ∃ X ( X ∈ A ) ⟹ [ ∀ Y ( Y ∈ A ⟺ Y ∈ B...

the properties of a Dedekind complete ordered field, meaning that any nonempty set of real numbers with an upper bound has a least upper bound. However...

have at most n nonempty rows), or the irreducible representations of the special linear group SLn (when they have at most n − 1 nonempty rows), or the...

property may be used to characterize realizers of finite partial orders: A nonempty set R of linear extensions is a realizer if and only if it reverses every...

family of disjoint measurable sets of finite positive measure. The only nonempty convex open set in L p ( [ 0 , 1 ] ) {\displaystyle L^{p}([0,1])} is the...

transversal of an undirected graph is a set of vertices of the graph that has a nonempty intersection with every odd cycle in the graph. Removing the vertices of...

space X {\displaystyle X} is compact if and only if every collection of nonempty closed subsets of X {\displaystyle X} with empty intersection admits a...

Relationships between the four types of propositions in the square of opposition (Black areas are empty, red areas are nonempty.)...

three and the geometric mean is always in between. (If all values in a nonempty data set are equal, the three means are always equal.) It is the special...

is sometimes also required. The Hasse diagram of the subset relation on nonempty intersections of maximal cliques forms an oriented tree. The convex subsets...

_{\geq 0}^{E}} ). A polymatroid on the ground set E {\displaystyle E} is a nonempty compact subset P {\displaystyle P} , the set of independent vectors, of...

least-upper-bound property for sets of real numbers, which states that every bounded, nonempty set of real numbers has a supremum. If the domain is the set of all real...

closed and convex. The Bondareva–Shapley theorem: the core of a game is nonempty if and only if the game is "balanced". Every Walrasian equilibrium has...

{\displaystyle {\mathcal {Y}}} and Y {\displaystyle {\mathcal {Y}}} have nonempty intersection. Such a triple ( G , X , Y ) {\displaystyle (G,{\mathcal {X}}...

numbers that can be obtained by adding up the elements of some finite nonempty subset of D. The set of all finite sums over D is often denoted as FS(D)...

conformal mappings in R2 can be much more complicated – for example, all nonempty proper simply connected planar domains are conformally equivalent, by the...

for every a in the domain of the relation. Every reflexive relation on a nonempty domain has infinite descending chains, because any constant sequence is...

domain of a function of several real variables is supposed to contain a nonempty open subset of ⁠ R n {\displaystyle \mathbb {R} ^{n}} ⁠. A real-valued...

in some groups. Let G {\displaystyle G} be a non-trivial group. Three nonempty subsets S , T , U ⊂ G {\displaystyle S,T,U\subset G} are said to have the...

denoted by juxtaposition. Then H is a subgroup of G if and only if H is nonempty and closed under products and inverses. Closed under products means that...

if its symmetric closure is anti-symmetric. A reflexive relation on a nonempty set X {\displaystyle X} can neither be irreflexive, nor asymmetric ( R...

x ) L ( y ) e . {\displaystyle x\cdot y=L(x)y+L(y)x-L(x)L(y)e.} X is a nonempty set and F a field. FX is the set of functions from X to F. If f, g are...

{\displaystyle G} defined on the class of all nonempty sets such that G ( x ) ∈ x {\displaystyle G(x)\in x} for every nonempty set x . {\displaystyle x.} This is...

orientation from S. Note that any compact, connected, oriented surface with nonempty boundary in Euclidean 3-space is the Seifert surface associated to its...

In mathematics, a filter on a set is a collection of nonempty subsets which is closed under taking supersets and finite intersections. An example of filter...

supporting hyperplane. Conversely, if S {\displaystyle S} is a closed set with nonempty interior such that every point on the boundary has a supporting hyperplane...

x',x\rangle \right\}.} By the Hahn–Banach theorem this set is nonempty. In the Hilbert space case (using the canonical duality between a Hilbert...

Hausdorff spaces is Hausdorff Connectedness The disjoint union of two or more nonempty topological spaces is disconnected product topology, the dual construction...

also occur in dynamical systems. A continuum C {\displaystyle C} is a nonempty compact connected metric space. The arc, the n-sphere, and the Hilbert...

with the additional property that, for any two simplices that have a nonempty intersection, their intersection is a vertex, edge, or higher dimensional...

infinite. Given a tuple ( P k ) k ∈ K {\displaystyle (P_{k})_{k\in K}} of nonempty subsets of X {\displaystyle \displaystyle {X}} , called the sites, a zone...

diagram universal quantification), red shaded regions with an x represent nonempty sets (to diagram existential quantification), and the other regions have...

element x in X is in exactly one of these subsets (i.e., the subsets are nonempty mutually disjoint sets). Equivalently, a family of sets P is a partition...

Alternative proofs use Helly's theorem or the Perron–Frobenius theorem. For any nonempty P ⊂ R d {\displaystyle P\subset \mathbb {R} ^{d}} , define its Carathéodory's...

structuring a head element and a list tail into a bigger list or destructuring a nonempty list into its head element and its tail. It does not denote "of type" as...

there are five that are essentially different. A null semigroup on any nonempty set with a chosen zero, or a left/right zero semigroup on any set. The...

_{i=1}^{n}A_{i}} is also an object. Furthermore, if fi is an object for i in an nonempty index set I implies the product ∏ i ∈ I f i : X → ∏ A i {\displaystyle...

is linear or projective, or if the intersection of all hyperplanes is nonempty, the intersection lattice is a geometric lattice. (This is why the semilattice...

\mathbb {R} :\mu (\{x:f(x)<b\})=0\}} if the set of essential lower bounds is nonempty, and as − ∞ {\displaystyle -\infty } otherwise; again there is an alternative...

generalizes both direct product and ultraproduct. Let {Si | i ∈ I} be a nonempty family of structures of the same signature σ indexed by a set I, and let...

subderivatives at x 0 {\displaystyle x_{0}} for a convex function is a nonempty closed interval [ a , b ] {\displaystyle [a,b]} , where a {\displaystyle...

defined as complete CAT(0) spaces. An Hadamard space is defined to be a nonempty complete metric space such that, given any points x {\displaystyle x} and...

disk, and an edge connecting each pair of circles or disks that have a nonempty intersection. Unit disk graphs may be formed in a different way from a...

the property that every two hyperedges in S have a nonempty intersection, then S itself has a nonempty intersection (a vertex that belongs to all hyperedges...

called its kernel. Families with empty kernel are called free; those with nonempty kernel, fixed. The empty set cannot belong to any family with the finite...

the Rådström embedding theorem, which implies that the collection of all nonempty compact convex subsets of a normed real vector-space (endowed with the...

R n {\displaystyle f:X\to \mathbb {R} ^{n}} , where the domain X is a nonempty subset of the Euclidean space R n {\displaystyle \mathbb {R} ^{n}} . ch(X)...

∩ D = S ∩ ( C ∩ D ) {\displaystyle (S\cap C)\cap D=S\cap (C\cap D)} is nonempty. Therefore, ( S ∩ C ) {\displaystyle (S\cap C)} must be stationary. See...

nonempty interior, then C* is pointed, i.e. C* contains no line in its entirety. If C is a cone and the closure of C is pointed, then C* has nonempty...

that, if the convex hulls of finitely many red and blue points have a nonempty intersection, then there exists a subset of d + 2 {\displaystyle d+2} points...

{\displaystyle I\in \mathbb {R} } , such that P ∩ I {\displaystyle P\cap I} is nonempty, and the restriction f | P ∩ I {\displaystyle f|_{P\cap I}} is continuous...

\circ } (X) is closed under finite intersections. X is sober, i.e., every nonempty irreducible closed subset of X has a unique generic point. From that X...

take any values for elements of Y not in the image. A function f with nonempty domain is injective if and only if it has a left inverse. An elementary...

Baire) if and only if the set of points at which is X is locally compact is nonempty (respectively dense) in X. Exhaustion by compact sets Lindelöf space –...

implies that f {\displaystyle f} attains its supremum and infimum on any (nonempty) compact set K {\displaystyle K} . Thus, we have the following generalization...

stating that there exist arbitrarily large sets of integers, all of whose nonempty sums belong to the same part. Using this definition, the only known Schur...

{\displaystyle S} s. However, the language it generates is simply the set of all nonempty strings consisting of a {\displaystyle a} s and/or b {\displaystyle b}...

assumption that every Cauchy sequence of real numbers converges. Let S be a nonempty set of real numbers. If S has exactly one element, then its only element...

examples and counterexamples. Closed sets have empty interior Given a nonempty open set A ⊆ X {\displaystyle A\subseteq X} every x ≠ p {\displaystyle...

true. The axiom of dependent choice can be stated as follows: For every nonempty set X {\displaystyle X} and every total relation R {\displaystyle R} on...

"even − even = even" forces 0 to be even is part of a general pattern: any nonempty subset of an additive group that is closed under subtraction must be a...

(Dedekind's Axiom) For every partition of all the points on a line into two nonempty sets such that no point of either lies between two points of the other...

Others provide a more permissive definition which allows fixed points. A nonempty subset S of X is a cycle of σ {\displaystyle \sigma } if the restriction...

locally path connected. X {\displaystyle X} is a scattered space, as each nonempty subset has a first element, which is an isolated element of the set. The...

increasing and continuous). More generally, if X {\displaystyle X} is a nonempty set and λ {\displaystyle \lambda } is a cardinal, then C ⊆ [ X ] λ {\displaystyle...

vertices are connected by edges whenever the corresponding sets have a nonempty intersection. When the sets are geometric objects, the result is a geometric...

any nonempty open subset U {\displaystyle U} of X . {\displaystyle X.} Expanding out the negation of density, it is equivalent that each nonempty open...

k ∈ K {\textstyle (P_{k})_{k\in K}} be a tuple (indexed collection) of nonempty subsets (the sites) in the space X {\textstyle X} . The Voronoi cell, or...

non-empty, then the nucleolus is in the ε-core. The least-core is the smallest nonempty ε-core (for the smallest ε for which the ε-core is non-empty). The nucleolus...

information theory, and other areas. Let X {\displaystyle {\mathcal {X}}} be a nonempty set, sometimes referred to as the index set. A symmetric function K : X...

\gamma } ⁠, contradicting irreflexivity. If ⁠ A {\displaystyle A} ⁠ is a nonempty set of ordinals, then ⁠ min A = ⋂ A {\displaystyle \min A=\bigcap A} ⁠...

{\mathit {CPS}}^{i},y\in {\mathit {PPS}},y\succeq \sum _{i}(x^{i}-r^{i})} Any nonempty subset of households may eliminate all other households, while retaining...

only if each of the closed axis-aligned orthants having p as apex has a nonempty intersection with K. The orthogonal convex hull is also known as the rectilinear...

topology of X {\displaystyle X} . In fact, if X {\displaystyle X} is a nonempty space, G ≃ H ∗ ( C ∗ ( X ; G ) ) {\displaystyle G\simeq H^{*}(C^{*}(X;G))}...

of opposition In the Venn diagrams black areas are empty and red areas are nonempty. The faded arrows and faded red areas apply in traditional logic....

are the same as nonempty sets in classical logic, it is not possible to produce a model in the classical sense that contains a nonempty set X {\displaystyle...

subsets of vertices A,B ⊆ V(G). In the equation, the minimum is over all nonempty sets S of at most n⁄2 vertices and ∂S is the edge boundary of S, i.e.,...

an injection is the identity relation. A partition of X is a set P of nonempty subsets of X, such that every element of X is an element of a single element...

{\displaystyle \infty } characteristic function ι C {\displaystyle \iota _{C}} of a nonempty, closed, convex set C {\displaystyle C} we have that prox ι C ⁡ ( x ) =...

The set of all elements q with qp = p is a compact semigroup, and is nonempty by the previous step, so by minimality it is the whole of S and therefore...

into N+ = N or {∞} such that κ(∅) = ∞ κ(A) = min {κ(w) | w ∈A} for each nonempty A ∈A; κ(A) is called the negative rank of A. One interpretation of k is...

leads to a state in A for each set Y in P for which X ∩ Y is nonempty and Y \ X is nonempty do replace Y in P by the two sets X ∩ Y and Y \ X if Y is in...

The subgroup test provides a necessary and sufficient condition for a nonempty subset ⁠ H {\displaystyle H} ⁠ of a group ⁠ G {\displaystyle G} ⁠ to be...

cannot be mapped one-to-one back into that set. That the set above is nonempty follows from Hartogs' theorem, which says that for any well-orderable cardinal...

{\mathbb {R} }^{d}} that are well-behaved, in the sense that a region is a nonempty compact subset that is the closure of its interior. We take a set of regions...

} {\displaystyle \{1,\ldots ,n\}} (that is, a collection of disjoint, nonempty subsets whose union is { 1 , … , n } {\displaystyle \{1,\ldots ,n\}} )...

the polytope A x ≤ b {\displaystyle A\mathbf {x} \leq \mathbf {b} } is nonempty’. To solve linear programmes using feasibility tests, one first checks...

injective map f : U → R m {\displaystyle f:U\to \mathbb {R} ^{m}} for a nonempty open set U ⊆ R n {\displaystyle U\subseteq \mathbb {R} ^{n}} . To see this...

vertices vi and vj by an edge whenever the corresponding two sets have a nonempty intersection, that is, E ( G ) = { { v i , v j } ∣ i ≠ j , S i ∩ S j ≠...

{\displaystyle \{N\subseteq M|P\not \in \operatorname {Ass} (N)\}} (which is nonempty since zero is in it). The set has a maximal element Q {\displaystyle Q}...

the frontier is determined by pairwise victories. The Landau set is a nonempty subset of the Smith set. It was first discovered by Nicholas Miller. The...

over K for the operations of V. Equivalently, a linear subspace of V is a nonempty subset W such that, whenever w1, w2 are elements of W and α, β are elements...

in which ZFC is typically formalized, the domain of discourse must be nonempty. Hence, it is a logical theorem of first-order logic that something exists –...

definition. For example, an irreflexive relation cannot be reflexive (on a nonempty domain set). except all xi are equal for all i beyond some n, for a reflexive...

(⁠7/9⁠, ⁠8/9⁠) of the upper third to { 0 | } = 1; and so forth. This maps a nonempty open interval of I onto each element of S∗, monotonically. The residue...

{\displaystyle X} over k {\displaystyle k} . Also, if U {\displaystyle U} is a nonempty open subset of X {\displaystyle X} , then dim ⁡ U = dim ⁡ X {\displaystyle...

on binary numbers. Let A {\displaystyle A} and B {\displaystyle B} be nonempty subsets of the set N {\displaystyle N} of nonnegative integers. The sumset...

the collection of equivalence classes of continuous paths from p to q is nonempty. In categorical terms, the assertion is that the objects p and q are in...

Given a set of integers, FIND-SUBSET-SUM is the problem of finding some nonempty subset of the integers that adds up to zero (or returning the empty set...

it has dense image. For such an f, if V is a nonempty open affine subset of Y, then there is a nonempty open affine subset U of X such that f(U) ⊂ V and...

the Poincaré–Bendixson theorem provides a simple characterization of all nonempty, compact ω {\displaystyle \omega } -limit sets that contain at most finitely...

the set of all its nonempty ground instances i.e. all strings resulting from consistent replacement of its variable symbols by nonempty strings of constant...

) = Mor ⁡ ( X , P ) {\displaystyle P(X)=\operatorname {Mor} (X,P)} is nonempty, i.e. the morphism f {\displaystyle f} admits at least a section s : X...

completely metrizable if and only if it is Gδ in X. A countable product of nonempty metrizable spaces is completely metrizable in the product topology if and...

{\displaystyle X} with the cocountable topology is Lindelöf, since every nonempty open set omits only countably many points of X {\displaystyle X} . It is...

property (i.e., each finite subset of F {\displaystyle F} has nonempty intersection) has nonempty intersection. Consider the set of all families of subsets...

\Theta } signifies an asymptotically tight bound. Yao showed that every nonempty set of nonnegative integers is the score set for some tournament. In social...

of a topological space X is a cardinal κ such that for every set of κ nonempty open subsets of X there is some point of X contained in κ of these subsets...

This makes it faster to discover that K is empty. However, when K is nonempty, there are examples in which the central-cut method finds a feasible point...

function to be "piecewise", the subdomains are typically required to be nonempty intervals (some may be degenerate intervals, i.e. single points or unbounded...

-primary ideals. If p {\displaystyle {\mathfrak {p}}} is maximal, a finite nonempty product of p {\displaystyle {\mathfrak {p}}} -primary ideals is p {\displaystyle...

of the set if and only if the add-tag list less the remove-tag list is nonempty. An optimization that eliminates the need for maintaining a tombstone set...

[X]^{(n)}=([X]\lor 0')^{(n-1)}} . The low basis theorem states that any nonempty Π 1 0 {\displaystyle \Pi _{1}^{0}} class in 2 ω {\displaystyle 2^{\omega...

for which c ∧ a ≤ b. In the finite case, every nonempty distributive lattice, in particular every nonempty finite chain, is automatically complete and completely...

interior has empty interior. Every meagre set has empty interior. Every nonempty open set is nonmeagre. Every comeagre set is dense. Whenever a countable...

π3 is the set of primes p such that a Sylow p-subgroup P has SCN3(P) nonempty and normalizes a nontrivial abelian subgroup of order prime to p. π4 is...

{\displaystyle h_{A}(x)=|x\cdot a|} . The support function of a compact nonempty convex set is real valued and continuous, but if the set is closed and...

set of maximizers C ∗ {\displaystyle C^{*}} is upper-hemicontinuous with nonempty and compact values. As a consequence, the sup {\displaystyle \sup } may...

the set of alternatives is finite) condition for a simple game to have a nonempty "core" (the set of socially "best" alternatives) for all profiles of individual...

{\displaystyle x} is an irrational number is nowhere cliquish, since every nonempty open set G {\displaystyle G} contains some y 1 , y 2 {\displaystyle y_{1}...

from the German Summe, meaning "sum") on a set X {\displaystyle X} is a nonempty collection Σ {\displaystyle \Sigma } of subsets of X {\displaystyle X}...

Demazure and Grothendieck showed that pinned reductive group schemes over any nonempty scheme S are classified by root data. This statement includes the existence...

convex balanced hull of A . {\displaystyle A.} The bipolar theorem: For any nonempty cone A {\displaystyle A} in some linear space X , {\displaystyle X,} the...

dimensions. Whenever each pair in a collection of these circles has a nonempty intersection, there exists an intersection point for the whole collection;...

Bieberbach shows that the image of an injective analytic mapping can exclude a nonempty open subset of C n {\displaystyle \mathbb {C} ^{n}} . The case of polynomials...

set. For example, in Zn with Euclidean metric, a sphere of radius r is nonempty only if r2 can be written as sum of n squares of integers. An octahedron...

space S is both hyperconnected (since every nonempty open set contains 1) and ultraconnected (since every nonempty closed set contains 0). It follows that...

a set of arcs connecting the states S is a nonempty subset of N that contains start states G is a nonempty subset of N that contains the goal states....

the domain of discourse or universe, which is usually required to be a nonempty set. For example, consider the sentence "There exists x such that x is...

is nonempty, the Cartesian product may be empty if the axiom of choice, which is equivalent to the statement that every such product is nonempty, is...

element. In this section internal sets are discussed in more detail. Every nonempty internal subset of *R that has an upper bound in *R has a least upper bound...

classical bivalent first-order logic, hence to truth and falsity under any (nonempty) universe of discourse. Hence the following were not logic for Quine: Higher-order...

(possibly infinite) poset P is equivalent to P being well-founded: every nonempty subset of P has a minimal element (also called the minimal condition or...

halfspaces must be nonempty. By Helly's theorem, it follows that the intersection of all of these halfspaces must also be nonempty. Any point in this...

homophone of" "is a co-worker of" "is a teammate of" By definition, a nonempty relation cannot be both symmetric and asymmetric (where if a is related...

w, so S can reject. If R rejects N, then the language accepted by N is nonempty, so M does halt on input w, so S can accept. Thus, if we had a decider...

form a Helly family: every set of translates that have nonempty pairwise intersections has a nonempty intersection. Moreover, these are the only convex bodies...

{\displaystyle S.} Proof As long as S ∖ { x } {\displaystyle S\setminus \{x\}} is nonempty, its closure will be X . {\displaystyle X.} It is only empty when S {\displaystyle...

in an initial heap and two players alternately divide a heap into two nonempty heaps of different sizes. Thus, six objects may be divided into piles of...

total order on R {\displaystyle \mathbb {R} } with the property that every nonempty subset of R {\displaystyle \mathbb {R} } has a least element in this ordering...

called a smallest element. In other words, if A {\displaystyle A} is a nonempty subset of the nonnegative integers, then there exists an element of A {\displaystyle...

=\{\varnothing ,X\},} since the preimage of any point in the range is some proper, nonempty subset of X , {\displaystyle X,} which is not an element of the trivial...

reduced homology groups of a contractible space are all trivial. For a nonempty topological space X the following are all equivalent: X is contractible...

widely accepted by mathematicians. It states that given a collection of nonempty sets there is a single set C that contains exactly one element from each...

of all the solvable ideals of g {\displaystyle {\mathfrak {g}}} . It is nonempty since { 0 } {\displaystyle \{0\}} is a solvable ideal, and it is a solvable...

of points such that every range of R {\displaystyle {\mathcal {R}}} has nonempty intersection with H {\displaystyle H} , i.e., is hit by H {\displaystyle...

∩ U j → G {\displaystyle t_{ij}:U_{i}\cap U_{j}\to G} defined on each nonempty overlap, such that the cocycle condition t i k ( x ) = t i j ( x ) t j...

as in surgery theory. Differential spaces Let M {\displaystyle M} be a nonempty set. Suppose that some family of real functions on M {\displaystyle M}...

In mathematics, an acyclic space is a nonempty topological space X in which cycles are always boundaries, in the sense of homology theory. This implies...

two vertices adjacent in L(H) when their corresponding hyperedges have a nonempty intersection in H. In other words, L(H) is the intersection graph of a...

statement that there exist arbitrarily large sets of integers all of whose nonempty sums are monochromatic, may be seen as a special case of Rado's theorem...

Space-Filling Curves". YouTube. Retrieved November 1, 2019. "Distribution of nonempty triangles inside a fractal rep-4-tile". The On-Line Encyclopedia of Integer...

set-like: R−1[x] = {y : y R x} is a set for every x, R is well-founded: every nonempty subset S of X contains an R-minimal element (i.e. an element x ∈ S such...

Grassmannian of p-planes in TxM. In cases of interest, Gx(φ) is always nonempty. Let G(φ) be the union of Gx(φ) for all x ∈ M {\displaystyle x\in M} ,...

(f)),} the interior of the effective domain of f , {\displaystyle f,} is nonempty, and that ϕ {\displaystyle \phi } is continuous on the set int ⁡ ( dom...

consider the set S of all proper ideals of R containing I. The set S is nonempty since I ∈ S. Furthermore, for any chain T of S, the union of the ideals...

theory, appearances to the contrary notwithstanding. Foundation: Each nonempty class A is disjoint from at least one of its members. ∀ A [ A ≠ ∅ → ∃ b...

modal operator ◊ {\displaystyle \Diamond } which is allowed to take any nonempty sequence of arguments. The arithmetical interpretation of ◊ ( p 1 , … ...

coalition with size ≥ 2 {\displaystyle \geq 2} . Partition the coalition into nonempty subsets G 1 , G 2 {\displaystyle G_{1},G_{2}} . Fix distinct x , y , z...

{\displaystyle H} , which is partially ordered by set inclusion. It is nonempty as it contains the empty set and thus by the maximal principle, it contains...

= ∅ {\displaystyle \gamma (0)=\emptyset } and so that, for each finite nonempty set A = { a 0 , … , a k } {\displaystyle A=\{a_{0},\ldots ,a_{k}\}} , γ...

sets, and so on. For example, the set containing only the empty set is a nonempty pure set. In modern set theory, it is common to restrict attention to the...

partitions of an n {\displaystyle n} -element set into k {\displaystyle k} nonempty subsets. A weak ordering may be obtained from such a partition by choosing...

where S X {\displaystyle SX} is the suspension of X {\displaystyle X} . If nonempty open sets U ⊂ R n {\displaystyle U\subset \mathbb {R} ^{n}} and V ⊂ R m...

Replacement into theorems (Potter 2004: §13.3). Let A and a denote sequences of nonempty sets, each indexed by n. Countable Choice: Given any sequence A, there...

Identifies the object's subtype. Object payload: Variable Optional payload. If nonempty, it contains a data structure, whose size is a multiple of 32 bits. ICMP...

{\displaystyle f} is constant. Suppose that D {\displaystyle D} is a bounded nonempty connected open subset of C {\displaystyle \mathbb {C} } . Let D ¯ {\displaystyle...

stated above, a first-order interpretation is usually required to specify a nonempty set as the domain of discourse. The reason for this requirement is to guarantee...

it is defined essentially the same as the first-order logic: There is a nonempty set M {\displaystyle M} , the universe of the interpretation. For each...

language L over the alphabet Σ = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, +, =}: Every nonempty string that does not contain "+" or "=" and does not start with "0" is...

section by a tangent plane). The paraboloid is elliptic if every other nonempty plane section is either an ellipse, or a single point (in the case of a...

theorem, which states the following: Wernicke's theorem: Assume G is planar, nonempty, has no faces bounded by two edges, and has minimum degree 5. Then G has...

} If Ω ⊊ R n {\displaystyle \Omega \subsetneq \mathbb {R} ^{n}} is an nonempty convex open set, then for 1 < p < ∞ {\displaystyle 1<p<\infty } and every...

a point of ωX is a family F {\displaystyle {\mathcal {F}}} of closed nonempty subsets of X such that F {\displaystyle {\mathcal {F}}} is closed under...

and b, while the only nonterminal is S. The language described is all nonempty strings of as and bs that end in a. This grammar is regular: no rule has...

that shares certain properties of the real line Continuum (topology), a nonempty compact connected metric space (sometimes Hausdorff space) Continuum hypothesis...

^{m}} and a polyhedral convex ordering cone C {\displaystyle C} having nonempty interior and containing no lines. The feasible set is S = { x ∈ R n : A...

belong to the nested set collection. Some authors do not assume that B is nonempty. The second condition states that the intersection of every couple of sets...

(1987). Setting φ(x) in Separation to x≠x, and assuming that the domain is nonempty, assures the existence of the empty set. Adjunction implies that if x is...

be an abelian group. If A {\displaystyle A} and B {\displaystyle B} are nonempty finite subsets of G {\displaystyle G} satisfying | A + B | < | A | + |...

inner product space X {\displaystyle X} and let U {\displaystyle U} be a nonempty subset of X {\displaystyle X} . A correspondence f : U ⇉ X {\displaystyle...

^{k}\to \Sigma ^{n}} . Here, Σ {\displaystyle \Sigma } is a finite and nonempty set and k {\displaystyle k} and n {\displaystyle n} are integers. The meaning...

satisfy the following maximum principle: if ⁠ K {\displaystyle K} ⁠ is a nonempty compact subset of ⁠ U {\displaystyle U} ⁠, then ⁠ f {\displaystyle f} ⁠...

nontotient, a sparsely totient number, and a Harshad number. There are 510 nonempty proper subsets of an 9-element set. It is the sum of eight consecutive...

{\displaystyle \Sigma } is compact, convex, and nonempty. r ( σ ) {\displaystyle r(\sigma )} is nonempty. r ( σ ) {\displaystyle r(\sigma )} is upper hemicontinuous...

if all non empty open subsets of X are dense, or if any two nonempty open sets have nonempty intersection. A subset F of a topological space X is called...

{p\,\mid \,x}{p\in \mathbb {P} }}D_{p}(x).} Let S {\displaystyle S} be a nonempty set of primes. The arithmetic subderivative of x {\displaystyle x} with...

Braces follow the Kernighan and Ritchie style ("Egyptian brackets") for nonempty blocks and block-like constructs Darwin, Ian F. (1988). Checking C programs...

algebra expression, denoting the Marked state, a primitive value. Hence a nonempty Cross is an operator, while an empty Cross is an operand because it denotes...

real number is often defined as a Dedekind cut. By definition, this is a nonempty proper lower subset of Q {\displaystyle \mathbb {Q} } with no maximal element...

times in Y then it occurs m + n times in U. A function d on the set of nonempty finite multisets of elements of a set M is a metric if d ( X ) = 0 {\displaystyle...

picture. Like partial orders and equivalence relations, preorders (on a nonempty set) are never asymmetric. A preorder can be visualized as a directed graph...

members of the incidence algebra are the functions f assigning to each nonempty interval [a, b] a scalar f(a, b), which is taken from the ring of scalars...

}} , where tk goes to infinity. It can be shown that ω(y) is compact, nonempty, and a union of flow lines. If z = lim k → ∞ Φ t k ∈ ω ( y ) , {\displaystyle...

that shares certain properties of the real line Continuum (topology), a nonempty compact connected metric space (sometimes a Hausdorff space) Continuum...

let s be a nonzero rational section of L (that is, a section on some nonempty open subset of L), which exists by local triviality of L. Define the Weil...

A cut of an undirected graph is a partition of the vertices into two nonempty subsets, the sides of the cut. The subset of edges that have one endpoint...

p.133), and is equivalent to various other statements: Every class of nonempty sets has a choice function. V \ {∅} has a choice function (where V is the...

{\displaystyle U.} Given a set X , {\displaystyle X,} a filter is a collection of nonempty subsets of X {\displaystyle X} that is closed under finite intersection...

holomorphic functions on a connected open subset U of Cn, if they coincide on a nonempty open subset of U, they agree on U. In terms of sheaf theory, the uniqueness...

corresponding productions {ananW1, ..., ananWn-1, anan, H}, where the Wk are nonempty words; he then proved the universality of a very small (4-state, 6-symbol)...

Let X be the set of all linearly independent subsets of V. The set X is nonempty since the empty set is an independent subset of V, and it is partially...

are equivalent to each other: The set is well ordered. That is, every nonempty subset has a least element. Transfinite induction works for the entire...

called biseparable, if one can find a partition of the parties in two nonempty disjoint subsets A {\displaystyle A} and B {\displaystyle B} with A ∪ B...

four. Removing these four vertices and their 12 incident edges leaves a nonempty set of edges, any of which together with the four removed vertices forms...

both convex and compact. By property (4), W ( A ) {\textstyle W(A)} is nonempty. Let z {\textstyle z} be a point on the boundary of W ( A ) {\textstyle...

removing some of the tokens and dividing the remaining tokens into two nonempty heaps. Heaps other than the selected heap remain unchanged. The last player...

of non-singular points consists of just the ideal (0) and contains no nonempty open sets. Excellent ring H. Matsumura, Commutative algebra ISBN 0-8053-7026-9...

just R as a module over itself, then Rn is an Mn(R)-module. If S is a nonempty set, M is a left R-module, and MS is the collection of all functions f :...

0-\infty } measures that are not zero measures. Let X {\displaystyle X} be nonempty, let A {\displaystyle {\cal {A}}} be a σ {\displaystyle \sigma } -algebra...

n {\displaystyle n\times n} matrix A is decomposable if there exists a nonempty proper subset I ⊂ { 1 , … , n } {\displaystyle I\subset \{1,\dots ,n\}}...

ineffable. Let P κ ( A ) {\displaystyle P_{\kappa }(A)} be the set of all nonempty subsets of A {\displaystyle A} which have cardinality < κ {\displaystyle...

{\displaystyle {\mathcal {U}}_{n}\subset \mathbb {R} ^{(n-1)^{2}}} has a nonempty interior. The unistochastic matrix corresponding to the unitary n × n {\displaystyle...

) {\displaystyle \gcd(p,q)} . Theorem—Let u , v {\displaystyle u,v} be nonempty words. If the infinite words u u u ⋯ {\displaystyle uuu\cdots }  and v...

(a+1,a,\ldots )\prec (b,b,\ldots )} Hence, there is a set of disjoint nonempty segments in ( X , ≺ ) {\displaystyle (X,\prec )} with a cardinality like...

distinct elements of P, then either x − y or y − x is in P. If S is any nonempty subset of P, then S + P = x + P for some x in C . {\displaystyle \mathbb...

functions f + g and f g are defined only if the domains of f and g have a nonempty intersection; in this case, their domain is the intersection of the domains...

is no corresponding axiom of intersection. If A {\displaystyle A} is a nonempty set containing E {\displaystyle E} , it is possible to form the intersection...

usual metric. A simple corollary of the theorem is that the Cantor set is nonempty, since it is defined as the intersection of a decreasing nested sequence...

multifunction on Ω {\displaystyle \Omega } taking values in the set of nonempty closed subsets of X {\displaystyle X} . Suppose that ψ {\displaystyle \psi...

the countable chain condition: every collection of mutually disjoint, nonempty open intervals in R is countable. In order theory, the famous Suslin problem...

denoted X < ω {\displaystyle X^{<\omega }} . With this notation, a tree is a nonempty subset T {\displaystyle T} of X < ω {\displaystyle X^{<\omega }} , such...

Jain & Nagpaul (1994), p. 196. Kay (2021), p. 44. The set still must be nonempty. Dummit & Foote (1999), p. 48 discuss this criterion written multiplicatively...

stack automaton is a tuple ⟨Q,Σ,Γ,δ,q0,Z0,F,[,],]⟩ where Q, Σ, and Γ is a nonempty finite set of states, input symbols, and stack symbols, respectively, [...

coming from formal language theory, is the free semigroup generated by a nonempty set (an alphabet), with string concatenation as the binary operation, and...

vertices are connected by edges whenever the corresponding sets have a nonempty intersection. Each vertex is represented as a set, and every two vertices...

characterization of the Sierpiński carpet. Recall that a continuum is a nonempty connected compact metric space. Suppose X is a continuum embedded in the...

Kleinian group is Schottky if and only if it is finitely generated, free, has nonempty domain of discontinuity, and all non-trivial elements are loxodromic. A...

+,\cdot ,<)} be an arbitrary ordered field, and M {\displaystyle M} a nonempty set; a function d : M × M → F + ∪ { 0 } {\displaystyle d:M\times M\to F^{+}\cup...

additional regularity of the cone (specifically, the dual cone A* needs to have nonempty interior). Reinhardt domain Siegel domain Some conventions instead define...

− ∞ , x ) , ( y , + ∞ ) {\displaystyle (-\infty ,x),(y,+\infty )} are nonempty since y ∈ ( − ∞ , x ) , x ∈ ( y , + ∞ ) {\displaystyle y\in (-\infty ,x)...

{\displaystyle z\mapsto |z|} on the complex numbers is not complex analytic on any nonempty open subset of C {\displaystyle \mathbb {C} } . Piecewise defined functions...

requirements for all d ≤ n: The requirement for d=−1 means that X should be nonempty. The requirement for d=0 means that X should be path-connected. The requirement...

theorem characterizing, in terms of A, the circumstances under which every nonempty X_b has a least element. In the special case where A contains all the rows...

alternating set x 0 , . . . x n {\displaystyle x_{0},...x_{n}} along with nonempty closed intervals I 0 , . . . I n {\displaystyle I_{0},...I_{n}} called...

generalizations of the hyperplane separation theorem, which states that two disjoint nonempty convex subsets of a finite-dimensional space R n {\displaystyle \mathbb...

Leray cover with respect to F {\displaystyle {\mathcal {F}}} if, for every nonempty finite set { i 1 , … , i n } {\displaystyle \{i_{1},\ldots ,i_{n}\}} of...

order of a bramble is the smallest size of a set of vertices that has a nonempty intersection with all of the subgraphs. The treewidth of a graph is the...

minimum points of g must then be two disjoint faces whose images have a nonempty intersection. This same general statement, when applied to a hypersphere...

{\displaystyle (\Leftarrow )} may fail unless C {\displaystyle C} has nonempty interior. Let C ⊆ H {\displaystyle C\subseteq H} be non-empty, closed,...

{\displaystyle \geq 3} , or surfaces with genus ≥ 2 {\displaystyle \geq 2} and nonempty boundary or at least two punctures, are not CAT(0). Some free-by-cyclic...

hereditarily normal. A totally ordered set is said to be complete if every nonempty subset that has an upper bound, has a least upper bound. For example, the...

every nonempty feasible set X contains an element x such that X ∖ { x } {\displaystyle X\setminus \{x\}} is feasible. This implies that any nonempty, finite...

\varphi } is convex, at each real number x {\displaystyle x} we have a nonempty set of subderivatives, which may be thought of as lines touching the graph...

choice,[better source needed] two other characterizations are possible: Any nonempty set S of submodules of the module has a maximal element (with respect to...

{\displaystyle f} is normal then f {\displaystyle f} commutes with suprema; for any nonempty set A {\displaystyle A} of ordinals, f ( sup A ) = sup f ( A ) = sup {...

of each color) in such a way that the convex hulls of these sets have a nonempty intersection. For instance, the two-dimensional case (proven by Bárány...

with the following properties: (B1) B {\displaystyle {\mathcal {B}}} is nonempty. (B2) If A {\displaystyle A} and B {\displaystyle B} are distinct members...

infimum is attained as the set of all such c {\displaystyle c} is closed, nonempty, and bounded from below. It is important to bear in mind that this operator...

y]\subseteq K.} If K {\displaystyle K} is a nonempty subset of X {\displaystyle X} and F {\displaystyle F} is a nonempty subset of K , {\displaystyle K,} then...

B {\displaystyle B} if and only if A × B {\displaystyle A\times B} has nonempty intersection with every entourage. Uniformly continuous maps will then...

compact groups and manifolds. Let X {\displaystyle X} be an arbitrary nonempty set and let G {\displaystyle G} be a commutative semigroup acting on X...

means that a total order can be imposed on its elements such that every nonempty subset has a first element with respect to that order. The order of a well-ordered...

System.out.println( new Lines(new DataInputStream(System.in)) .takeWhile(nonEmpty) .map(fun(String s) -> int { return Integer.parseInt(s); }) .reduceLeft(0...

{\mathcal {N}}(X)} from a metric space X {\displaystyle X} to the family of nonempty subsets of X {\displaystyle X} is said to be Lipschitz if there exists...

{\displaystyle F_{\vee }(X)} may be realised as the set of all finite nonempty subsets of X {\displaystyle X} , with ordinary set union as the join operation...

subset V of An is called an affine algebraic set if V = Z(S) for some S. A nonempty affine algebraic set V is called irreducible if it cannot be written as...

queues are active) while GPS serves infinitesimal amounts of data from each nonempty queue and offers this part on any interval. If the queues have packets...

{\mathcal {F}}} . If we take the axiom of choice to be true then every nonempty family of finite character has a maximal element with respect to inclusion...

this way has the Helly property, because any subfamily of C with pairwise nonempty intersections must correspond to a clique in G, which can be extended to...

either spatially close or descriptively close. Spatially close sets have nonempty intersection. In other words, spatially close sets are not disjoint sets...

intersection theorem, the intersection of all the elements in any filter is nonempty. If G {\displaystyle G} is a filter intersecting the dense set D n {\displaystyle...

association (with teachings).[citation needed] "Empty" should not be asserted."Nonempty" should not be asserted. Neither both nor neither should be asserted. They...

pseudocompact. In a countably compact space, every locally finite family of nonempty subsets is finite. Every countably compact paracompact space is compact...

topological vector space V {\textstyle V} is an AR; more generally, every nonempty convex subset of such a vector space V {\textstyle V} is an AR. For example...

generate a subring of the center. Let R be a ring. A left ideal of R is a nonempty subset I of R such that for any x, y in I and r in R, the elements x +...

a metric space, and let A {\displaystyle A} and B {\displaystyle B} be nonempty compact subsets of S {\displaystyle S} . The distance from a point a {\displaystyle...

admitted, M8 asserts the existence of the fusion of all members of any nonempty set. Any mereological system in which M8 holds is called general, and its...

either an ascent or a descent. An ascending run of a permutation is a nonempty increasing contiguous subsequence that cannot be extended at either end;...

\Delta } . During each phase, the algorithm removes all nodes of the first nonempty bucket and relaxes all outgoing edges of weight at most Δ {\displaystyle...

element), the assumption that X {\displaystyle X} is nonempty is redundant and because of that, the nonempty-ness assumption is sometimes omitted from the statement...

and are closely related to tangent cones. Let S {\displaystyle S} be a nonempty subset of a real normed vector space ( X , ‖ ⋅ ‖ ) {\displaystyle (X,\|\cdot...

theorem, gives necessary and sufficient conditions for a game to have nonempty core. Facility location (optimization problem) Facility location (competitive...

Then, S is the cartesian product of all these simplices. It is indeed a nonempty, compact and convex subset of Rkn. The function φ(x) associates with each...

n ≥ 1 and let μ denote the Lebesgue measure on Rn. Let A and B be two nonempty compact subsets of Rn. Then the following inequality holds: [ μ ( A + B...

of X . {\displaystyle X.} That is, F {\displaystyle {\mathcal {F}}} is nonempty and is closed under complement and under unions and intersections of countably...

follows: If G is acyclic, then r(G) = 0. If G is strongly connected and E is nonempty, then r ( G ) = 1 + min v ∈ V r ( G − v ) , {\displaystyle r(G)=1+\min...

topology. This space is not T0. It is limit point compact because every nonempty subset has a limit point. An example of T0 space that is limit point compact...

set S can be recovered from the IFS by fixed point iteration: for any nonempty compact initial set A0, the iteration Ak+1 = H(Ak) converges to S. The...

definitions. A conditionally complete lattice is a lattice in which every nonempty subset that has an upper bound has a join (that is, a least upper bound)...

the general definition of a self-concordant function. Let C be a convex nonempty open set in Rn. Let f be a function that is three-times continuously differentiable...

of NFU. To see that weak Extensionality holds is straightforward: each nonempty element of V j ( α ) + 1 {\displaystyle V_{j(\alpha )+1}} inherits a unique...

the one-point set is a generator for the category of sets. In fact, any nonempty set is a generator. In the category of sets, any set with at least two...

b+L is an affine plane), and K is a closed pointed convex cone with a nonempty interior. Every convex program can be converted to the conic form. To use...

can be expressed logically as the statement that there does not exist a nonempty subset C {\displaystyle C} of T {\displaystyle T} for which each vertex...

For instance, in a graph with a Hamiltonian cycle, the removal of any nonempty set of vertices partitions the cycle into at most as many pieces as the...

argument shows that P ( X ) T {\displaystyle {\mathcal {P}}(X)^{T}} is nonempty. It is also compact and convex in the weak-* topology; hence by the Krein–Milman...

function F σ {\displaystyle F_{\sigma }} is either meager or else has nonempty interior. This is not always equivalent with countability. The integral...

if for every p ∈ P there is some q ∈ D with q ≤ p. A filter on P is a nonempty subset F of P such that if p < q and p ∈ F then q ∈ F, and if p ∈ F and...

substring of the form w w w {\displaystyle www} with w {\displaystyle w} some nonempty finite string. It is not known whether every string appearing in the sequence...

of its normal subgroups and their quotient groups. A Lyndon word is a nonempty string of symbols that is smaller, lexicographically, than any of its cyclic...

continuous real-valued function on a closed interval has a maximum, and thus a nonempty argmax . {\displaystyle \operatorname {argmax} .} Argument of a function...

\Theta \subset \mathbb {R} ^{k}} . Then If Θ {\displaystyle \Theta } has nonempty interior in R k {\displaystyle \mathbb {R} ^{k}} , then given any IID samples...

path connected subspaces U1, U2. Suppose U1 ∩ U2 is path connected and nonempty, and let x0 be a point in U1 ∩ U2 that will be used as the base of all...

topology that is T1 but is not Hausdorff (T2). This follows since no two nonempty open sets of the cofinite topology are disjoint. Specifically, let X {\displaystyle...

in Ci and the product g1…gn is trivial. Then A is called rigid if it is nonempty, G acts transitively on it by conjugation, and each element of A generates...

1 , 0 ) {\displaystyle [-1,0)} and ( 0 , 1 ] {\displaystyle (0,1]} are nonempty, disjoint, and open in the subspace topology on Y {\displaystyle Y} , and...

subgroups has order greater than 2. Suppose there exist pairwise disjoint nonempty subsets X1, X2, ...,Xk of X such that the following holds: For any i ≠...

Dedekind Complete (DC) if every nonempty set, bounded above, has a supremum; Super Dedekind Complete (SDC) if every nonempty set, bounded above, has a countable...

"multiplicatively closed subset" is appropriate. A filter on a set X is a nonempty collection of nonempty subsets of X that is closed under finite intersection and under...

{\mathcal {C}},\mu )} be an ordered triple where X {\textstyle X} is a nonempty set, C {\textstyle {\mathcal {C}}} is a ring of sets, and μ {\textstyle...

at least one element of the set F (a HALT state) for the language to be nonempty. Q = { A , B , C , HALT } ; {\displaystyle Q=\{A,B,C,{\text{HALT}}\};}...

equal to the smallest element in the intersection. This shows that every nonempty intersection of a finite number of arithmetic progressions is again an...

{\displaystyle H} is a subset S {\displaystyle S} of X {\displaystyle X} that has nonempty intersection with each hyperedge. Blocking sets are sometimes also called...

x\in U.} (Here V {\displaystyle V} intersects S {\displaystyle S} means nonempty intersection V ∩ S ≠ ∅ {\displaystyle V\cap S\neq \varnothing } ). If a...

contour sets at A are topologically open; If the lower countour set of A is nonempty, then A is in its closure. A similar approach was taken by Richter. Therefore...

both concavity of the log-likelihood function and compactness of some (nonempty) upper level sets of the log-likelihood function, or existence of a compact...

{\displaystyle \Phi (x)} as well as to types. Infinity. There exists a nonempty binary relation R {\displaystyle R} over the individuals of the lowest...

sequence is empty. For f : S × S → S, define a new function Fl on finite nonempty sequences of elements of S, where F l ( a 0 , k ) = { a 0 , k = 1 f ( F...

topological space is said to be connected if it is not the union of two disjoint nonempty open sets. A set is open if it contains no point lying on its boundary;...

The word enumerable is used because the following are equivalent for a nonempty subset B of the natural numbers: B is the domain of a computable function...

\circ } f ∘ {\displaystyle \circ } f) n, as is used in Church numerals. A nonempty list can be represented by a Church pair, where first contains the list's...

vertices vi and vj by an edge whenever the corresponding two sets have a nonempty intersection. That is, the edge set of G is E ( G ) = { ( v i , v j ) ∣...

G:g(x)=0\}} . Then E {\displaystyle E} is closed in G {\displaystyle G} and nonempty. It is open too: for every x ∈ E {\displaystyle x\in E} , | g ( y ) | =...

universal cone from lim F to F. If the index category J is connected (and nonempty) then the unit of the adjunction is an isomorphism so that lim is a left...

are adjacent in G; hence u is dominated by i. Second, since D must be nonempty, each i ∈ I is adjacent to a vertex in D. Conversely, let D be a dominating...

utility maximum always exists, Marshallian demand correspondence must be nonempty at every value that corresponds with the standard budget set. x ∗ ( p ...

attains the value − ∞ {\displaystyle -\infty } and its effective domain is nonempty. This means that there exists some x ∈ X {\displaystyle x\in X} at which...

for any nonempty open set U ⊆ X {\displaystyle U\subseteq X} , the interior of f ( U ) {\displaystyle f(U)} in Y {\displaystyle Y} is nonempty. Such a...

subset V of G. The intersection U ∩ V {\displaystyle U\cap V} is then nonempty but then this implies a is in the image of f 1 {\displaystyle f_{1}} ....

theorem claims that if X {\displaystyle X} is common knowledge at some nonempty set of states, then all the numbers x a {\displaystyle x_{a}} are the same...

topological space X {\displaystyle X} is hyperconnected if and only if every nonempty open set is dense in X . {\displaystyle X.} A topological space is submaximal...

strongly connected or strong if and only if it cannot be partitioned into two nonempty subsets L {\displaystyle L} and W {\displaystyle W} of losers and winners...

with the property that every d + 1 sets have a nonempty intersection, then the whole family has a nonempty intersection. Helly families, named after this...

\forall i\in I\colon \lambda _{i}<\kappa \right\}.} That the set above is nonempty comes from the fact that κ = ⋃ i ∈ κ { i } {\displaystyle \kappa =\bigcup...

I,f,g)} , where Γ {\displaystyle \Gamma } and I {\displaystyle I} are nonempty abstract sets. f {\displaystyle f} is the write function: f : Γ ∗ × I →...

positive integers, and let F {\displaystyle {\mathcal {F}}} be an arbitrary nonempty family of subsets of the universe { 1 , … , n } {\displaystyle \{1,\dots...

question, Zermelo states that a necessary and sufficient condition is the nonemptyness of a certain set, containing all possible sequences of moves such that...

crossing is counted whenever two edges that are disjoint in the graph have a nonempty intersection in the plane. The question is then, what is the minimum number...

Lj ⊆ Li. Symmetrically, C is said to satisfy the MFF-condition if every nonempty finite set D has at most finitely many minimal concepts in C. Finally,...

R^{m}.} In the special case that the set : Y ( x ) {\displaystyle Y(x)} is nonempty for all x ∈ X {\displaystyle x\in X} GSIP can be cast as bilevel programs...

a nonempty subset of X. If x is a point in X, the distance of x from A is defined as d(x, A) = inf{ d(x, a): a in A}. If A and B are both nonempty subsets...

base of closed compact neighborhoods. The space is ultraconnected, as any nonempty closed set contains the point p . {\displaystyle p.} Therefore the space...

Ran(T) ∩ {\displaystyle \cap } span { v 1 } {\displaystyle \{v_{1}\}} is nonempty, i.e. there exists some complex number α such that T w = α v 1 {\displaystyle...

{\displaystyle \rho _{E}(x)=\min\{n\geq 1:f^{n}(x)\in E\}} if this set is nonempty, and otherwise, let it be ρ E ( x ) = ∞ {\displaystyle \rho _{E}(x)=\infty...

explicitly or implicitly, that as a topological space a surface is also nonempty, second-countable, and Hausdorff. It is also often assumed that the surfaces...

Duggan–Schwartz theorem A result about voting systems designed to choose a nonempty set of winners from the preferences of certain individuals, where each...

is more general than σ {\displaystyle \sigma } . More formally, take a nonempty infinite set V {\displaystyle V} of auxiliary variables such that no equation...

definition of J i . {\displaystyle J_{i}.} So, Σ {\displaystyle \Sigma } is a (nonempty) partially ordered set by set inclusion such that every chain in Σ {\displaystyle...

, the space of ω {\displaystyle \omega } -tame J {\displaystyle J} is nonempty and contractible. Gromov used this theory to prove a non-squeezing theorem...

Borel algebra of X whose support is X. This means that μ(U) > 0 for any nonempty open subset U of X. A recent generalization[citation needed] replaces these...

interesting case is when the stable locus X s {\displaystyle X^{s}} is nonempty; X s {\displaystyle X^{s}} is the open set of semistable points that have...

in Rn and a number t, decide whether cTx ≤ t for all x in K. Strong nonemptyness problem (SNEMPT): Decide whether K is empty, and if not, find a point...

a certain set. The main part of the proof will consider the case of a nonempty set, and examine the members in detail; in the case where the set is empty...

collection of theorems. The least-upper-bound property states that every nonempty subset of real numbers having an upper bound (or bounded above) must have...

by infinitely many hypercubes, he labeled as δn. The convex hull of any nonempty subset of the n + 1 points that define an n-simplex is called a face of...

line which, initially, is at the left of all input segments.) While Q is nonempty, find and remove the event from Q associated with a point p with minimum...

E\setminus \bigcup _{j\neq i}I_{j}} , where each of the sets on the right is nonempty by the inductive hypothesis. We can assume z i ∈ I i {\displaystyle z_{i}\in...

numbers that contains x {\displaystyle x} . Then, every number maps to a nonempty set and no number maps to the empty set. But the empty set is a member...

weight μ {\displaystyle \mu } . Indeed, since the last box on the first nonempty line of the skew diagram can only contain an entry 1, the entire first...

but they can differ on more general sets. Let A {\displaystyle A} be a nonempty closed subset of the Banach space X {\displaystyle X} . The Clarke's tangent...

{ 1 } {\displaystyle f:\emptyset \to \{1\}} from the empty poset to a nonempty poset has no retract, because there is no order-preserving map g : { 1...

\limits _{x\in X}\{g(x)=f(x)+E[Q(x,\xi )]\}} Here X {\displaystyle X} is a nonempty closed subset of R n {\displaystyle \mathbb {R} ^{n}} , ξ {\displaystyle...

then the projection is unique if it exists; if W {\displaystyle W} is nonempty, closed, and convex and Ω ⊂ R n {\displaystyle \Omega \subset \mathbb {R}...

non-trivial coordinate subspace means a linear subspace spanned by any nonempty proper subset of standard basis vectors of Fn. More explicitly, for any...

random points selected with respect to a uniform distribution inside K. The nonempty intersection of half-spaces in R d {\displaystyle \mathbb {R} ^{d}} . The...

lemma is proven by repeatedly finding large sets of clauses that have a nonempty common intersection in a given formula, and replacing the formula by two...

a lattice L is a sublattice of L that is also a convex set of L. Every nonempty convex sublattice can be uniquely represented as the intersection of a...