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 $r\in ℝ$ 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 $f:X\to ℝ$ 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 $Z(f-|f|)$.

9. If $C(X)$ is considered as an algebra over $ℝ$, 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 $ℝ$. However, this definition turns out to be equivalent to the one given for $C(X)$, by the observation that $Z(f)=Z(|f|\wedge \widehat{1})$.