properties of the closure operator

Suppose X is a topological spaceMathworldPlanetmath, and let A¯ be the closureMathworldPlanetmathPlanetmath of A in X. Then the following properties hold:

  1. 1.

    A¯=AA where A is the derived set of A.

  2. 2.

    AA¯, and A=A¯ if and only if A is closed

  3. 3.

    A¯= if and only if A=.

  4. 4.

    If Y is another topological space, then f:XY is a continuous mapMathworldPlanetmath, if and only if f(A¯)f(A)¯ for all AX. If f is also a homeomorphism, then f(A¯)=f(A)¯.

Title properties of the closure operator
Canonical name PropertiesOfTheClosureOperator
Date of creation 2013-03-22 15:17:05
Last modified on 2013-03-22 15:17:05
Owner matte (1858)
Last modified by matte (1858)
Numerical id 11
Author matte (1858)
Entry type Theorem
Classification msc 54A99