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.

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

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.

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.

$\{x\in X\mid 0\leq f(x)\}$ 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 $r\neq 0$ .

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).

Title	zero set of a topological space
Canonical name	ZeroSetOfATopologicalSpace
Date of creation	2013-03-22 16:56:06
Last modified on	2013-03-22 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