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The Cantor set is an unusual closed set in the sense that it consists entirely of boundary points and is nowhere dense. Singleton points (and thus finite sets) are closed in T 1 spaces and Hausdorff spaces. The set of integers is an infinite and unbounded closed set in the real numbers.
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 ...
The closure of a subset is the result of a closure operator applied to the subset. The closure of a subset under some operations is the smallest superset that is closed under these operations. It is often called the span (for example linear span) or the generated set.
Closure operators are determined by their closed sets, i.e., by the sets of the form cl(X), since the closure cl(X) of a set X is the smallest closed set containing X. Such families of "closed sets" are sometimes called closure systems or "Moore families". [1] A set together with a closure operator on it is sometimes called a closure space.
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. Closure The closure of a set is the smallest closed set containing the original set. It is equal to the intersection of all closed sets which contain it.
Abstract closure or interior operators can be used to define a generalized topology on the lattice. Since neither unions nor the empty set appear in the requirement for a Moore closure operator, the definition may be adapted to define an abstract unary operator : on an arbitrary poset.
The interval (,] = (,) [,] is a locally closed subset of . For another example, consider the relative interior of a closed disk in . It is locally closed since it is an intersection of the closed disk and an open ball.
In point-set topology, Kuratowski's closure-complement problem asks for the largest number of distinct sets obtainable by repeatedly applying the set operations of closure and complement to a given starting subset of a topological space. The answer is 14. This result was first published by Kazimierz Kuratowski in 1922. [1]