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?