completely separated

Proposition 1.

Let A,B be two subsets of a topological spaceMathworldPlanetmath X. The following are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath:

  1. 1.

    There is a continuous functionMathworldPlanetmathPlanetmath f:X[0,1] such that f(A)=0 and f(B)=1,

  2. 2.

    There is a continuous function g:X such that g(A)r<sg(B), r,s.


Clearly 1 implies 2 (by setting r=0 and s=1). To see that 2 implies 1, first take the transformation


so that h(A)0<1h(B). Then take the transformation f(x)=(h(x)0)1, where 0(x)=0 and 1(x)=1 for all xX. Then f(A)=(h(A)0)1=01=0 and f(B)=(h(B)0)1=h(B)1=1. Here, and denote the binary operationsMathworldPlanetmath of taking the maximum and minimum of two given real numbers (see ring of continuous functions for more detail). Since during each transformation, the resulting function remains continuous, the first assertion is proved. ∎

Definition. Any two sets A,B in a topological space X satisfying the above equivalent conditions are said to be completely separated. When A and B are completely separated, we also say that {A,B} is completely separated.

Clearly, two sets that are completely separated are disjoint, and in fact separated.

Remark. A T1 topological space in which every pair of disjoint closed setsPlanetmathPlanetmath are completely separated is a normal spaceMathworldPlanetmath. A T0 topological space in which every pair consisting of a closed set and a singleton is completely separated is a completely regular space.

Title completely separated
Canonical name CompletelySeparated
Date of creation 2013-03-22 16:54:51
Last modified on 2013-03-22 16:54:51
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 7
Author CWoo (3771)
Entry type Definition
Classification msc 54-00
Classification msc 54D05
Classification msc 54D15
Synonym functionally distinguishable