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 $f\widehat{r}$, where $\widehat{r}$ is the constant function valued at $r$.

3.
$Z(\widehat{r})=X$ iff $r=0$. Otherwise, $Z(\widehat{r})=\mathrm{\varnothing}$. In fact, $Z(f)=\mathrm{\varnothing}$ iff $f$ is a unit in the ring $C(X)$.

4.
Since $f(a)=0$ iff $$ for all $n\in \mathbb{N}$, and each $$ is open in $X$, we see that
$$ This shows every zero set is a ${G}_{\delta}$ (http://planetmath.org/G_deltaSet) set.

5.
For any $f\in C(X)$, $Z(f)=Z({f}^{n})=Z(f)$, where $n$ is any positive integer.

6.
$Z(fg)=Z(f)\cup Z(g)$.

7.
$Z(f)\cap Z(g)=Z({f}^{2}+{g}^{2})=Z(f+g)$.

8.
$\{x\in X\mid 0\le f(x)\}$ 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 $r\ne 0$.
The complement^{} of a zero set is called a cozero set. In other words, a cozero set looks like $\{x\in X\mid f(x)\ne 0\}$ for some $f\in C(X)$. By the last property above, a cozero set also has the form $$ 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 $Z(f)=Z(f\wedge \widehat{1})$.
References
 1 L. Gillman, M. Jerison: Rings of Continuous Functions, Van Nostrand, (1960).
Title  zero set of a topological space 

Canonical name  ZeroSetOfATopologicalSpace 
Date of creation  20130322 16:56:06 
Last modified on  20130322 16:56:06 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  10 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 54C50 
Classification  msc 54C40 
Classification  msc 54C35 
Defines  zero set 
Defines  level set 
Defines  cozero set 