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First of all Polish spaces are completely-metrizable, separable topological space and by zero-dimensional Polish space I mean that the Polish space has a (countable) basis made of clopen sets. It is clear that a zero-dimensional Polish space is totally disconnected, I was wondering whether also the convers holds true.

  • If we have a totally disconnected Polish space, is it also zero-dimensional (i.e. has a countable basis of clopen sets)? If not, is there a counter-example?

I think it would suffice to prove that every open set in the space includes a clopen set (clopen w.r.t the overall space). Total disconnection implies that every non-empty open set (not singleton) is disconnected, hence it contains a clopen set w.r.t the open set (i.e. its relative topology), which is not, in general, clopen w.r.t the overall space.

Since I've read it in my professor's notes, I'm prone to think that the thesis is true, but I'm having some problems in proving it. Some help?

Thanks

Lorenzo
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1 Answers1

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It seems that there are totally disconnected Polish spaces that are not zero dimensional.

Consider the standard Hilbert space $l^2$ of all square-summable real sequences. Let $X\subseteq l^2$ be the set of all sequences with irrational coefficients. $X$ is clearly separable. It is not hard to show that it is totally disconnected as well. It is also completely metrizable since it is a $G_\delta$ subset of $l^2$.

Now the "On homogeneous totally disconnected 1-dimensional spaces" paper (by Kazuhiro Kawamura, Lex G. Oversteegen and E. D. Tymchatyn) claims that $X$ is of dimension $1$.

freakish
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    In Jan van Mill's book "Infinite-dimensional topology, an introduction and prerequisites" there is a construction of an infinite-dimensional (!) Polish space that is totally disconnected. So we cannot even bound the dimension. Your example (complete Erdös space) is a well-known example, indeed $1$-dimensional like the standard Erdös space (all rational sequences in $\ell^2$), but which is not Polish. – Henno Brandsma May 06 '20 at 12:06
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    Also, this example is infinite-dimensional but only contains subspaces that are either zero-dimensional or infinite-dimensional, nothing inbetween. It's a weird object, like an indecomposable continuum is weird.. @Lorenzo – Henno Brandsma May 06 '20 at 12:14