closure
The closure ˉA of a subset A of a topological space X is the intersection of all closed sets containing A.
Equivalently, ˉA consists of A together with all limit points of A in X or equivalently x∈ˉA if and only if every neighborhood of x intersects A. Sometimes the notation cl(A) is used.
If it is not clear, which topological space is used, one writes ˉAX. Note that if Y is a subspace of X, then ˉAX may not be the same as ˉAY. For example, if X=ℝ, Y=(0,1) and A=(0,1), then ˉAX=[0,1] while ˉAY=(0,1).
Title | closure |
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Canonical name | Closure |
Date of creation | 2013-03-22 12:05:40 |
Last modified on | 2013-03-22 12:05:40 |
Owner | mathwizard (128) |
Last modified by | mathwizard (128) |
Numerical id | 9 |
Author | mathwizard (128) |
Entry type | Definition |
Classification | msc 54A99 |
Related topic | ClosureAxioms |
Related topic | Interior |