completely separated

Proposition 1.

Let $A\mathrm{,}B$ be two subsets of a topological space $X$. The following are equivalent:

1. There is a continuous function $f:X\to [0,1]$ such that $f(A)=0$ and $f(B)=1$,

2. There is a continuous function $g:X\to ℝ$ such that , $r,s\in ℝ$.

Proof.

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

$$h(x)=\frac{g(x)-r}{s-r}$$

so that . Then take the transformation $f(x)=(h(x)\vee 0)\wedge 1$, where $0(x)=0$ and $1(x)=1$ for all $x\in X$. Then $f(A)=(h(A)\vee 0)\wedge 1=0\wedge 1=0$ and $f(B)=(h(B)\vee 0)\wedge 1=h(B)\wedge 1=1$. Here, $\vee$ and $\wedge$ denote the binary operations 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 sets are completely separated is a normal space. 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