The question is also posted here.

The paper is Mizokami : On characterizations of spaces with $G_\delta$-diagonals

See its Theorem 1, also you can see the picture . http://picpaste.com/a-eaiF4d3t.bmp.

Theorem 1:A space $X$ has a $G_\delta$-diagonal iff there is an open mapping (single valued) $f$ from a metric space $T$ onto $X$ such that $$d(f^{-1}(p),f^{-1}(q))>0,$$ for distinct points $p, q \in X.$

The author difines $T$ as follows:

$T=\lbrace (\alpha_1,\alpha_2,...)\in N(A): \bigcap \lbrace U_{\alpha_n}^n: n\in N\rbrace\not=\emptyset \rbrace$, where $\lbrace \mathcal U_n=\lbrace U_{\alpha}^n: \alpha \in A, n \in N\rbrace$ is a sequence of open covering of $X$ satisfying the condition in Lemma 1. (it can be seen in the paper.)

The author difines $f: T \rightarrow X$ as follows:

$f(\alpha)=\bigcap \lbrace U_{\alpha_n}^n: n\in N \rbrace$ for $\alpha \in T$

My question is this:

1) **What is the topology the author used which make $T$ is metrizable?**

2) **Is $f$ continuous?**

Thanks for your help.

`Baire’s zero-dimensional metric space N(A).`

Baire space ( en.wikipedia.org/wiki/Baire_space_%28set_theory%29 ) is well-known in the descriptive set theory, it has a basis consisting of sets of all sequences with prescribed first $n$ elements and we can get a metric by putting $d(x,y)=1/\min\{n; x_n\ne y_n\}$. I have seen it so far only for sequences of integers, but using other set $A$ instead of integers probably does not make much difference. $\endgroup$ – Martin Sleziak Apr 26 '13 at 8:05