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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 $x\longmapsto min \{{|f(x)|, 1}\}$ ?

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What is $C^*(X)$? –  1015 Jan 29 '13 at 16:24
    
@julien: continuous and bounded functions. –  Martin Jan 29 '13 at 16:26
    
The set of all continuous, bounded, real -valued functions on a topological space $X$. –  TXC Jan 29 '13 at 16:27

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up vote 1 down vote accepted

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 $g \colon X \to [0,1]$.

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Basically, all you need is a continuous function $h \colon \mathbb{R} \to [0,1]$ such that $h(x) = 0$ iff $x = 0$ and set $g(x) = h(f(x))$. –  Martin Jan 29 '13 at 16:39

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