Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I meet a difficult in the Example 2.17 of this paper, which said:"It was observed by G.M. Reed that X is a Moore space and that X is continuously symmetrizable (see the details in [2]), and therefore, X has a zero-set diagonal and a regular $G_\delta$-diagonal. " I can't understand this sentence.

Could someone help me to understand it? What's the relation betwen "continuously symmetrizable" and the "diagonal property"?

Thanks ahead:)

share|cite|improve this question
up vote 2 down vote accepted

Well, from your previous question and its excellent answer by Scott, you know what continuously symmetrizable is. In particular we have a $d: X \times X \rightarrow \mathbb{R}$, where $d$ is continuous and $d(x,y) = 0$ iff $x = y$, and this means that $\Delta_X = \{ (x,x) \mid x \in X \}$, the diagonal of $X$ in $X \times X$, can be seen as $d^{-1}[\{0\}]$, and this means by definition that $\Delta_X$ is a zeroset in $X \times X$.

A subset $A \subset Y$, $Y$ a topological space, is called a regular $G_\delta$ iff $A = \cap_{n \in \mathbb{N}} \overline{O_n}$, where the $O_n$ are open sets containing $A$. So in particular such a set is closed (intersection of closed sets) and a $G_\delta$ (as $A = \cap_{n \in \mathbb{N}} O_n$ as well).

Now, it's not too hard to see that a zero-set is a regular $G_\delta$: in the above notation, let $f: A \rightarrow \mathbb{R}$ is continuous and $A = f^{-1}[\{0\}]$. Then define $O_n = f^{-1}[(-\frac{1}{n}, \frac{1}{n})]$ which are open supersets of $A$, and $\overline{O_n} \subset f^{-1}[[-\frac{1}{n}, \frac{1}{n}]]$, and so their intersection equals exactly $A = f^{-1}[\{0\}]$ (it holds in $\mathbb{R}$ and intersection commutes with $f^{-1}[.]$), so $A$ is a regular $G_\delta$. As $\Delta_X$ is a zero-set this shows the second part of the remark in the paper.

All these properties like having a regular $G_\delta$ diagonal, or stronger zero-set diagonal, or even stronger symmetrizable, are examples of so-called generalized metric properties (all metric spaces have them, and they have some sort of metric like structure), which are quite old and well-studied. See this article by Gruenhage for a nice overview by one of the more well-known researchers in this area.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.