Is it possible to define a zero-set of $X$ to be the zero-set of some $f\in C^{*}(X)$?

Question

It is possible to define a cozero-set of $X$ to be the cozero-set of some $f\in C^{*}(X)$, in fact;

Every cozero-set in $X$ is the cozero-set of a function taking values in $[0, 1]$.

$proof$: Given $f:X\longrightarrow \mathbb{R}$, consider the function $x\longmapsto min \{{|f(x)|, 1}\}$. This is continuous if $f$ is, takes values in $[0, 1]$, and has the same cozero- set as $f$.

Is it possible to define a zero-set of $X$ to be the zero-set of some $f\in C^{*}(X)$ ? Can this problem be solved with this function $x\longmapsto min \{{|f(x)|, 1}\}$ ?

Answer 1

Yes, this works. Alternatively, consider $g(x) = \frac{2}{\pi}|\arctan{f(x)}|$. This has the same zero-set and cozero-set as $f$ and is plainly a continuous function $g \colon X \to [0,1]$.