# 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. 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. 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. 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. 4.

Since $f(a)=0$ iff $|f(a)|<\frac{1}{n}$ for all $n\in\mathbb{N}$, and each $\{x\in X\mid|f(x)|<\frac{1}{n}\}$ is open in $X$, we see that

 $Z(f)=\bigcap_{n=1}^{\infty}\{x\in X\mid|f(x)|<\frac{1}{n}\}.$

This shows every zero set is a $G_{\delta}$ (http://planetmath.org/G_deltaSet) set.

5. 5.

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

6. 6.

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

7. 7.

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

8. 8.

$\{x\in X\mid 0\leq f(x)\}$ is a zero set, since it is equal to $Z(f-|f|)$.

9. 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 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).
Title zero set of a topological space ZeroSetOfATopologicalSpace 2013-03-22 16:56:06 2013-03-22 16:56:06 CWoo (3771) CWoo (3771) 10 CWoo (3771) Definition msc 54C50 msc 54C40 msc 54C35 zero set level set cozero set