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[parent] zero set of a topological space (Definition)

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 $ \vert f(a)\vert<\frac{1}{n}$ for all $ n\in \mathbb{N}$, and each $ \lbrace x\in X \mid \vert f(x)\vert<\frac{1}{n} \rbrace$ is open in $ X$, we see that
    $\displaystyle Z(f)=\bigcap_{n=1}^{\infty}\lbrace x\in X \mid \vert f(x)\vert<\frac{1}{n} \rbrace.$
    This shows every zero set is a $ G_{\delta}$ set.
  5. For any $ f\in C(X)$, $ Z(f)=Z(f^n)=Z(\vert f\vert)$, 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(\vert f\vert+\vert g\vert)$.
  8. $ \lbrace x\in X\mid 0\le f(x)\rbrace$ is a zero set, since it is equal to $ Z(f-\vert f\vert)$.
  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.

Bibliography

1
L. Gillman, M. Jerison: Rings of Continuous Functions, Van Nostrand, (1960).



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Also defines:  zero set, level set, cozero set

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Cross-references: equivalent, continuous functions, bounded, subring, countable, operations, intersection, union, closed under, complement, algebra, integer, positive, open, ring, unit, iff, constant function, metric, closed set, metric space, converse, closed, properties, subset, ring of continuous functions, topological space
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This is version 7 of zero set of a topological space, born on 2007-04-16, modified 2007-05-14.
Object id is 9201, canonical name is ZeroSetOfATopologicalSpace.
Accessed 1621 times total.

Classification:
AMS MSC: 54C35 (General topology :: Maps and general types of spaces defined by maps :: Function spaces)
 54C40 (General topology :: Maps and general types of spaces defined by maps :: Algebraic properties of function spaces)
 54C50 (General topology :: Maps and general types of spaces defined by maps :: Special sets defined by functions)
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