we shall say that a subspace $S$ of $X$ is $C$-embedded ($C^{*}$-embedded) in $X$ if every function in $C(S)$ ($ C^{*}(S) $) can be extended to a function in $C(X)$($C^{*}(X)$).
Zero - set means:
$ Z ( f ) = f^{-1 } ( \{ 0 \} ) = \{ x \in X \vert f ( x ) = 0 \}$
My questions:
1: let $S \subset X $, if every zero-set in $S$ is a zero-set in $X$ , is $S$, $C^{*}$-embedded in $X$?
2: A discrete zero-set is $C^{*}$-embedded if and only if all of its subsets are zero-sets?