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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.
The definition of a point of closure of a set is closely related to the definition of a limit point of a set.The difference between the two definitions is subtle but important – namely, in the definition of a limit point of a set , every neighbourhood of must contain a point of other than itself, i.e., each neighbourhood of obviously has but it also must have a point of that is not equal to ...
Every closed ball is a closed set in the topology induced on M by d. Note that the closed ball D(x; r) might not be equal to the closure of the open ball B(x; r). Closed set A set is closed if its complement is a member of the topology. Closed function A function from one space to another is closed if the image of every closed set is closed ...
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 counterintuitive, as the common meanings of open and closed are antonyms, but their mathematical definitions are not mutually exclusive .
Let be the set of integers again, and using the definition of from the previous example, define a subbase of open sets for any integer to be = {, +} if is an even number, and = {,} if is odd. Then the basis of the topology are given by finite intersections of the subbasic sets: given a finite set A , {\displaystyle A,} the open sets of X ...
set-valued function with a closed graph. If F : X → 2 Y is a set-valued function between topological spaces X and Y then the following are equivalent: F has a closed graph (in X × Y); (definition) the graph of F is a closed subset of X × Y; and if Y is compact and Hausdorff then we may add to this list:
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The interior of a closed subset of is a regular open subset of and likewise, the closure of an open subset of is a regular closed subset of . [2] The intersection (but not necessarily the union) of two regular open sets is a regular open set. Similarly, the union (but not necessarily the intersection) of two regular closed sets is a regular ...