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[parent] completely separated (Definition)
Proposition 1   Let $ A,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 $ g(A)\le r<s \le g(B)$, $ 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
$\displaystyle h(x)=\frac{g(x)-r}{s-r}$
so that $ h(A)\le 0< 1\le h(B)$. 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. $ \qedsymbol$

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 $ \lbrace A,B\rbrace$ 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.



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Cross-references: completely regular space, singleton, T0, normal space, closed sets, T1, separated, disjoint, equivalent, function, ring of continuous functions, real numbers, binary operations, transformation, implies, continuous function, the following are equivalent, topological space, subsets
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This is version 4 of completely separated, born on 2007-04-10, modified 2007-04-19.
Object id is 9175, canonical name is CompletelySeparated.
Accessed 879 times total.

Classification:
AMS MSC: 54-00 (General topology :: General reference works )
 54D05 (General topology :: Fairly general properties :: Connected and locally connected spaces )
 54D15 (General topology :: Fairly general properties :: Higher separation axioms )
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