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

Equivalently, A¯ consists of A together with all limit pointsPlanetmathPlanetmath of A in X or equivalently xA¯ 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 A¯X. Note that if Y is a subspaceMathworldPlanetmathPlanetmath of X, then A¯X may not be the same as A¯Y. For example, if X=, Y=(0,1) and A=(0,1), then A¯X=[0,1] while A¯Y=(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