$C$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 realvalued continuous function^{} $f:A\to \mathbb{R}$, there is a realvalued continuous function $g:X\to \mathbb{R}$ 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.
References
 1 L. Gillman, M. Jerison: Rings of Continuous Functions, Van Nostrand, (1960).
Title  $C$embedding 

Canonical name  Cembedding 
Date of creation  20130322 16:57:37 
Last modified on  20130322 16:57:37 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  6 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 54C45 
Synonym  Cembedded 
Synonym  C embedded 
Synonym  C*embedded 
Synonym  C* embedded 
Defines  $C$embedded 
Defines  ${C}^{*}$embedded 