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closure


The closureMathworldPlanetmathPlanetmath ˉA of a subset A of a topological spaceMathworldPlanetmath X is the intersectionMathworldPlanetmath of all closed setsPlanetmathPlanetmath containing A.

Equivalently, ˉA consists of A together with all limit pointsPlanetmathPlanetmath of A in X or equivalently xˉA if and only if every neighborhoodMathworldPlanetmathPlanetmath 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 subspaceMathworldPlanetmathPlanetmath of X, then ˉAX may not be the same as ˉAY. For example, if X=, Y=(0,1) and A=(0,1), then ˉAX=[0,1] while ˉAY=(0,1).

Title closure
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