The set $ C= C ( X ) $ of all continuous, real valued functions on a topological space$ X$ will be provided with an algebraic structure and an order struture.The subset $C^{*} = C^{*} (X)$ of $C(X)$ , consisting of all bounded function in $C(X)$.
A Zero-set means: $ Z ( f ) = f^{-1 } ( \{ 0 \} ) = \{ x \in X \vert f ( x ) = 0 \}$ , $ f \in C (X) $
Does anyone help me to prove the following problems?
1: Is every $C^{*}$ -embedded zero - set, $C$-embedded?
2:Is a subset $S$ of $\mathbb{R}$ $C$-embedded ($C^{*}$-embedded$ )if and only if it is closed?
3:Is $C(S)$ a homomorphism image of $C(X)$, if a (nonempty) subset $S$ of $X$ is $C$ - embedded in $X$?
1) I don’t know an answer, but the following criterion may be helpful for you. I recall that two subsets $A$ and $B$ of a topological space are called completely separated if there exists a continuous function $f:X\to [0,1]$ such that $f(x)=0$ for each $x\in A$ and $f(x)=1$ for each $x\in B$. According to this question, a $C^*$-embedded subset $Y$ of space $X$ is $C$-embedded in $X$ iff $Y$ is completely separated from every zero-set in $X$ disjoint from it.
2) Each $C^*$-embedded subset of $\Bbb R$ is closed by this answer. Conversely, each closed subset of $\Bbb R$ (or, more generally, of an arbitrary normal space) is $C$-embedded by Tietze-Urysohn extension theorem (see below its proof from Engelkings’ “General topology”).
3) It is easy to check that for each subset $S$ of the space $X$ the restriction map $r:f\mapsto f|S$ preserves all four arithmetic operations and if $f,g\in C(X)$ and $f\le g$, that is $f(x)\le g(x)$ for each point $x\in X$ then $f|S\le g|S$, so the map $r_S$ should be a homomorphism. Moreover, the map $r_S$ is sujective iff the set $S$ is $C$-embedded in $X$.