$C$-embedding

Let $X$ be a topological space, and $C(X)$ the ring of continuous functions on $X$. A subspace $A\subseteq X$ is said to be $C$-embedded (in $X$) if every function in $C(A)$ can be extended to a function in $C(X)$. More precisely, for every real-valued continuous function $f:A\to ℝ$, there is a real-valued continuous function $g:X\to ℝ$ such that $g(x)=f(x)$ for all $x\in A$.

If $A\subseteq X$ is $C$-embedded, $f\mapsto g$ (defined above) is an embedding of $C(A)$ into $C(X)$ by axiom of choice, and hence the nomenclature.

Similarly, one may define $C^{\mathrm{*}}$-embedding on subspaces of a topological space. Recall that for a topological space $X$, $C^{*}(X)$ is the ring of bounded continuous functions on $X$. A subspace $A\subseteq X$ is said to be $C^{\mathrm{*}}$-embedded (in $X$) if every $f\in C^{*}(A)$ can be extended to some $g\in C^{*}(X)$.

Remarks. Let $A$ be a subspace of $X$.

1. If $A$ is $C$-embedded in $X$, and $A\subseteq Y\subseteq X$, then $A$ is $C$-embedded in $Y$. This is also true for $C^{*}$-embeddedness.

2. If $A$ is $C$-embedded, then $A$ is $C^{*}$-embedded: for if $f$ is a bounded continuous function on $A$, say $-n\le f\le n$, and $g$ is its continuous extension on $X$, then $-n\vee (g\wedge n)$ is a bounded continuous extension of $f$ on $X$.

3. The converse, however, is not true in general. A necessary and sufficient condition that a $C^{*}$-embedded set $A$ is $C$-embedded is:

   if a zero set is disjoint from $A$, the it is completely separated from $A$.

   Since any pair of disjoint zero sets are completely separated, we have that if $A$ is a $C^{*}$-embedded zero set, then $A$ is $C$-embedded.

Title	$C$-embedding
Canonical name	Cembedding
Date of creation	2013-03-22 16:57:37
Last modified on	2013-03-22 16:57:37
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	6
Author	CWoo (3771)
Entry type	Definition
Classification	msc 54C45
Synonym	C-embedded
Synonym	C embedded
Synonym	C*-embedded
Synonym	C* embedded
Defines	$C$-embedded
Defines	$C^{*}$-embedded