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 $f^{-1}(r):=\lbrace x\in X\mid f(x)=r\rbrace$ 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. $Z(\hat{r})=X$ iff $r=0$ Otherwise, $Z(\hat{r})=\varnothing$ In fact, $Z(f)=\varnothing$ iff $f$ is a unit in the ring $C(X)$
4. Since $f(a)=0$ iff $|f(a)|<\frac{1}{n}$ for all $n\in \mathbb{N}$ and each $\lbrace x\in X \mid |f(x)|<\frac{1}{n} \rbrace$ is open in $X$ we see that $$Z(f)=\bigcap_{n=1}^{\infty}\lbrace x\in X \mid |f(x)|<\frac{1}{n} \rbrace.$$ This shows every zero set is a <</A>124#>$G_{\delta}$ http://planetmath.org/encyclopedia/G_deltaSet.html 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. $\lbrace x\in X\mid 0\le f(x)\rbrace$ is a zero set, since it is equal to $Z(f-|f|)$
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 $\lbrace x\in X\mid f(x)\ne 0\rbrace$ for some $f\in C(X)$ By the last property above, a cozero set also has the form $\operatorname{pos}(f):=\lbrace x\in X\mid 0<f(x)\rbrace$ 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 $A$ $Z(A):=\lbrace Z(f)\mid f\in A\rbrace$ 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 $Z(f)=Z(|f|\wedge \hat{1})$

Bibliography

1

L. Gillman, M. Jerison: Rings of Continuous Functions, Van Nostrand, (1960).