Definition Suppose A and B are subsets of a topological spaceMathworldPlanetmath X. Then A and B are separated provided that


where A¯ is the closure operatorPlanetmathPlanetmathPlanetmath ( in X.


  1. 1.

    If A,B are separated in X, and f:XY is a homeomorphismPlanetmathPlanetmath, then f(A) and f(B) are separated in Y.


  1. 1.

    On , the intervals (0,1) and (1,2) are separated.

  2. 2.

    If d(x,y)r+s, then the open ballsPlanetmathPlanetmath Br(x) and Bs(y) are separated (proof.) (

  3. 3.

    If A is a clopen set, then A and A are separated. This follows since S¯=S when S is a closed setPlanetmathPlanetmath.


The above definition follows [1]. In [2], separated sets are called strongly disjoint sets.


  • 1 J.L. Kelley, General Topology, D. van Nostrand Company, Inc., 1955.
  • 2 G.J. Jameson, Topology and Normed Spaces, Chapman and Hall, 1974.
Title separated
Canonical name Separated
Date of creation 2013-03-22 15:16:34
Last modified on 2013-03-22 15:16:34
Owner matte (1858)
Last modified by matte (1858)
Numerical id 15
Author matte (1858)
Entry type Definition
Classification msc 54-00
Classification msc 54D05