of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all...

a closed set. A set may be both open and closed (a clopen set). The empty set and the full space are examples of sets that are both open and closed. A...

In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed. That this is possible may seem...

In abstract algebra, a multiplicatively closed set (or multiplicative set) is a subset S of a ring R such that the following two conditions hold: 1 ∈...

union-closed sets conjecture, also known as Frankl’s conjecture, is an open problem in combinatorics posed by Péter Frankl in 1979. A family of sets is said...

the union of S and its boundary, and also as the intersection of all closed sets containing S. Intuitively, the closure can be thought of as all the points...

open sets in X are open, or equivalently, if arbitrary unions of closed sets are closed, or, again equivalently, if the open sets are the upper sets of...

{\displaystyle Y.} Likewise, a closed map is a function that maps closed sets to closed sets. A map may be open, closed, both, or neither; in particular...

In mathematics, a subset of a given set is closed under an operation on the larger set if performing that operation on members of the subset always produces...

locally closed if any of the following equivalent conditions are satisfied: E {\displaystyle E} is the intersection of an open set and a closed set in X...

topology, as every algebraic set is closed for the usual topology. The generalization of the Zariski topology to the set of prime ideals of a commutative...

of a set X is the smallest closed set containing X. Such families of "closed sets" are sometimes called closure systems or "Moore families". A set together...

Look up closed in Wiktionary, the free dictionary. Closed may refer to: Closure (mathematics), a set, along with operations, for which applying those operations...

characterization. In the space X , {\displaystyle X,} every open set is k-open and every closed set is k-closed. The space X {\displaystyle X} together with the new...

mathematics, a Borel set is any subset of a topological space that can be formed from its open sets (or, equivalently, from closed sets) through the operations...

In mathematics, an upper set (also called an upward closed set, an upset, or an isotone set in X) of a partially ordered set ( X , ≤ ) {\displaystyle...

some cs-closed subset B {\displaystyle B} of X × Y {\displaystyle X\times Y} Every cs-closed set is lower cs-closed and every lower cs-closed set is lower...

open set and every non-degenerate closed interval (that is, a closed interval containing at least two distinct points) is a regular closed set. A singleton...

{\displaystyle {\mathcal {C}}} of closed sets forms a base for the closed sets if and only if for each closed set A {\displaystyle A} and each point...

particularly in mathematical logic and set theory, a club set is a subset of a limit ordinal that is closed under the order topology, and is unbounded...

countable, closed set is a set of uniqueness, a discovery which led him to the development of set theory. Paul Cohen, another innovator in set theory, started...

spaces is that each compact set is a closed set. For non-Hausdorff spaces, it can be that each compact set is a closed set (for example, the cocountable...

{\displaystyle C_{n}=T_{L}(C_{n-1})\cup T_{R}(C_{n-1}),} the explicit closed formulas for the Cantor set are C = [ 0 , 1 ] ∖ ⋃ n = 0 ∞ ⋃ k = 0 3 n − 1 ( 3 k + 1 3...

open). A subset of X may be open, closed, both (clopen set), or neither. The empty set and X itself are always both closed and open. A base (or basis) B for...

a boundaryless bounded set, while the half plane is unbounded yet has a boundary. A bounded set is not necessarily a closed set and vice versa. For example...

Case Closed, also known as Detective Conan (Japanese: 名探偵コナン, Hepburn: Meitantei Konan; lit. 'Great Detective Conan'), is a Japanese manga series written...

often used with closed sets (as in the normal separation axiom). The sets A {\displaystyle A} and B {\displaystyle B} are separated by closed neighbourhoods...

mathematics, a normal space is a topological space in which any two disjoint closed sets have disjoint open neighborhoods. Such spaces need not be Hausdorff in...

mathematics, an Fσ set (said F-sigma set) is a countable union of closed sets. The notation originated in French with F for fermé (French: closed) and σ for somme...

distinct points in X are separated. Equivalently, every single-point set is a closed set. Thus, X is T1 if and only if it is both T0 and R0. (Although one...