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 $g:X\to \mathbb{R}$ such that $$, $r,s\in \mathbb{R}$.
Proof.
Clearly 1 implies 2 (by setting $r=0$ and $s=1$). To see that 2 implies 1, first take the transformation
$$h(x)=\frac{g(x)r}{sr}$$ 
so that $$. 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.
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  20130322 16:54:51 
Last modified on  20130322 16:54:51 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  7 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 5400 
Classification  msc 54D05 
Classification  msc 54D15 
Synonym  functionally distinguishable 