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zero set of a topological space
Let $X$ be a topological space and $f\in C(X)$, the ring of continuous functions on $X$. The level set of $f$ at $r\in\mathbb{R}$ is the set $f^{{1}}(r):=\{x\in X\mid f(x)=r\}$. The zero set of $f$ is defined to be the level set of $f$ at $0$. The zero set of $f$ is denoted by $Z(f)$. A subset $A$ of $X$ is called a zero set of $X$ if $A=Z(f)$ for some $f\in C(X)$.
Properties. Let $X$ be a topological space and, unless otherwise specified, $f\in C(X)$.
1. Any zero set of $X$ is closed. The converse is not true. However, if $X$ is a metric space, then any closed set $A$ is a zero set: simply define $f:X\to\mathbb{R}$ by $f(x):=d(x,A)$ where $d$ is the metric on $X$.
2. The level set of $f$ at $r$ is the zero set of $f\hat{r}$, where $\hat{r}$ is the constant function valued at $r$.
3. 4. Since $f(a)=0$ iff $f(a)<\frac{1}{n}$ for all $n\in\mathbb{N}$, and each $\{x\in X\midf(x)<\frac{1}{n}\}$ is open in $X$, we see that
$Z(f)=\bigcap_{{n=1}}^{{\infty}}\{x\in X\midf(x)<\frac{1}{n}\}.$ This shows every zero set is a $G_{{\delta}}$ set.
5. 6. $Z(fg)=Z(f)\cup Z(g)$.
7. $Z(f)\cap Z(g)=Z(f^{2}+g^{2})=Z(f+g)$.
8. $\{x\in X\mid 0\leq f(x)\}$ is a zero set, since it is equal to $Z(ff)$.
9. If $C(X)$ is considered as an algebra over $\mathbb{R}$, then $Z(rf)=Z(f)$ iff $r\neq 0$.
The complement of a zero set is called a cozero set. In other words, a cozero set looks like $\{x\in X\mid f(x)\neq 0\}$ for some $f\in C(X)$. By the last property above, a cozero set also has the form $\operatorname{pos}(f):=\{x\in X\mid 0<f(x)\}$ for some $f\in C(X)$.
Let $A$ be a subset of $C(X)$. The zero set of $A$ is defined as the set of all zero sets of elements of $A$: $Z(A):=\{Z(f)\mid f\in A\}$. When $A=C(X)$, we also write $Z(X):=Z(C(X))$ and call it the family of zero sets of $X$. Evidently, $Z(X)$ is a subset of the family of all closed $G_{{\delta}}$ sets of $X$.
Remarks.

By properties 6. and 7. above, $Z(X)$ is closed under set union and set intersection operations. It can be shown that $Z(X)$ is also closed under countable intersections.

It is also possible to define a zero set of $X$ to be the zero set of some $f\in C^{*}(X)$, the subring of $C(X)$ consisting of the bounded continuous functions into $\mathbb{R}$. However, this definition turns out to be equivalent to the one given for $C(X)$, by the observation that $Z(f)=Z(f\wedge\hat{1})$.
References
 1 L. Gillman, M. Jerison: Rings of Continuous Functions, Van Nostrand, (1960).
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