This article is about the complement of an open set. For a set closed under an operation, see closure (mathematics). For other uses, see Closed (disambiguation).
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set.[1][2] In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a closed set is a set which is closed under the limit operation.
This should not be confused with a closed manifold.
^Rudin, Walter (1976). Principles of Mathematical Analysis. McGraw-Hill. ISBN 0-07-054235-X.
^Munkres, James R. (2000). Topology (2nd ed.). Prentice Hall. ISBN 0-13-181629-2.
of mathematics, a closedset is a set whose complement is an open set. In a topological space, a closedset can be defined as a set which contains all...
In abstract algebra, a multiplicatively closedset (or multiplicative set) is a subset S of a ring R such that the following two conditions hold: 1 ∈...
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...
a closedset. 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. Open...
Look up closed in Wiktionary, the free dictionary. Closed may refer to: Closure (mathematics), a set, along with operations, for which applying those operations...
In mathematics, an upper set (also called an upward closedset, an upset, or an isotone set in X) of a partially ordered set ( X , ≤ ) {\displaystyle...
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 closedset containing X. Such families of "closedsets" are sometimes called closure systems or "Moore families". A set together...
{\displaystyle Y.} Likewise, a closed map is a function that maps closedsets to closedsets. A map may be open, closed, both, or neither; in particular...
open sets in X are open, or equivalently, if arbitrary unions of closedsets are closed, or, again equivalently, if the open sets are the upper sets of...
Case Closed, also known as Detective Conan (Japanese: 名探偵コナン, Hepburn: Meitantei Konan, lit. 'Great Detective Conan'), is a Japanese detective manga series...
a boundaryless bounded set, while the half plane is unbounded yet has a boundary. A bounded set is not necessarily a closedset and vice versa. For example...
mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closedsets) through the operations...
functional analysis, a closed linear span of a set of vectors is the minimal closedset which contains the linear span of that set. Suppose that X is a...
used for closed manifold if the usual definition for manifold is used. The notion of a closed manifold is unrelated to that of a closedset. A line is...
locally closed if any of the following equivalent conditions are satisfied: E {\displaystyle E} is the intersection of an open set and a closedset in X...
Fatou set of a rational map can be classified into one of four different classes. J ( f ) {\displaystyle \operatorname {J} (f)} is the smallest closed set...
a closedset itself (as happens, for instance, if X {\displaystyle X} is a finite set or more generally a compact set), then it equals the closed convex...
and is the underpinning of concepts such as closedset and topological closure. Indeed, a set is closed if and only if it contains all of its limit points...
open sets become axioms defining closedsets: The empty set and X {\displaystyle X} are closed. The intersection of any collection of closedsets is also...
{\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...
is closed in X . {\displaystyle X.} The sum of a compact set and a closedset is closed. However, the sum of two closed subsets may fail to be closed (see...
Global Information