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I have a Problem in general, given some some Topological Space $(X, \tau)$ from which I know it is metrisable, how can I find a metric (that is at best in some sence constructive and easy, at the very best could be easily programmed on a Computer).

It came up to me because I am trying to construct metrics for Refinements of the Cantor-Space. The Cantorspace is the sets of all one-sided-infinite words over $X = \{ 0,1 \}$, i.e. the set $X^{\mathbb N}$. The Standard-Topology is given by the Basis $\mathcal B = \{ w \cdot X^{\mathbb N} : w \in X^{*} \} \cup \{ \emptyset \}$ and is also the Product-Topology on $X^{\mathbb N}$ where $X$ is equipped with the discrete topology. A metric which induces this Topology could be easily found, it is $$ \operatorname{d}(u,v) = 2^{-r} ~ \textrm{ with } ~ r = \min \{ n : u_n \ne v_n \} $$ with the convention $\min \emptyset = \infty$ and $2^{-\infty} = 0$. Details could be found for example in the Book Infinite Words by Perrin/Pin (see here).

Ok, now one refinement of the above Topology is given by the base $$ \mathcal B_A := \{ F : F\subseteq X^{\mathbb N} \land F \textrm{is a regular $\omega$-language and closed in Cantor space} \}. $$ Details on these Topology and the form of the closure and interior operator for it could be found in the Paper Topologies Refining the Cantor Topology on $X^{\mathbb N}$

Now the Space $X^{\mathbb N}$ equipped with the Topology generated by $\mathcal B_A$ is normal and second-countable, so according to Urysohn Metrisation Theorem is metrisable. So I have gone through the proof of Urysohn's Lemma and Urysohns Metrisation Theorem. And despite it's "constructive" charakter I am unable to derive any usable metric for the space with Baisis $\mathcal B_A$ from it.

Now at this point, does anybody know examples where something similar is desired or could point me to papers which could help me? Even some hints if the Urysohn functions look simpler on these spaces?

To get more specific, for the construcition of the Urysohn function I looked at Munkres, Topology. And there it is constructed by successively selecting open sets which are nested in each other, I have no idea how to derive a simple representation of such a function form this. Furthermore, I looked at the original paper Zum Metrisationsproblem. In it he starts by enumerating the Basis sets $$ \{ U_1, U_2, U_3, \ldots \} $$ and then selecting those pairs $\pi_n = (U_i, U_k)$ with $\overline{U_i} \subseteq U_k$, guess it would help me to give a simple characterisation of these sets, but I didn't succeeded. Then for each $n$ and pair $\pi_n$ he defines the continuous Urysohn function $$ f_n(x) = 0 ~ \textrm{ on } \overline{U_i}, \quad f_n(x) = 1 ~ \textrm{ on } X \setminus U_k. $$

I would be really glad for hints or references on how I could tackle this Problem of constructing easy and handy metrics, thanks!

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  • $\begingroup$ I would expect that one can find a particular metric for your space using the venerable "hawk eye" method, by which you float in the air until you spot the right gadget and descend onto it before it gets away. $\endgroup$ – Andrej Bauer Jul 28 '13 at 20:29
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Some aspects of the standard metrization theorems are non-constructive, so there is probably no general solution for the problem you pose.

However, for your space with a basis consisting of closed regular ω-languages, there is a simpler way since this is a zero-dimensional space. A second-countable Hausdorff zero-dimensional space $E$ with clopen basis $(B_i)_{i\in\mathbb{N}}$ can always be embedded into Cantor space via the map $f:E \to 2^{\mathbb{N}}$ where each $f(x)$ is the characteristic function of the set $\{i \in \mathbb{N} : x \in B_i\}$ (i.e. $f(x)(i) = 1$ iff $x \in B_i$). Then, $d(f(x),f(y))$, where $d$ is the usual metric on Cantor space, gives a metric for $E$.

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