(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let B = {{x} : x is in R\{pi}}U{R\K : K is a finite subset of R and pi is not in K}U{0} be a family of subsets of R. One needs to prove that B is a basis of some topology U and that (R, U) is not metrisable.

3. The attempt at a solution

Now, I managed to prove that B is a basis, but I'm not really sure about the metrisation issue.

With the material covered in the lectures I'm working with, I couldn't find a way to prove it, so I did some research and found the Nagata-Smirnov theorem which states that a space is metrisable iff it is regular and has a countably locally finite basis.

Now, I assumed that (R, U) is metrisable, hoping to arrive at a contradiction. It is not hard to prove that it is regular, so the only thing left is the countably locally finite basis issue. So, assume B is a union of a countable family of locally finite collections of subsets of R. Let x be some element in R. Then x has a neighbourhood which intersects countably many subsets of B (I'm not really sure about hsi conclusion). But it seems that every neighbourhood of x intersects B in an uncountable number of sets, i.e. whatever the neighbourhood looks like, it always intersects an uncountable number of the sets R\K (I'm not sure about this, either).

I really need a push here, thanka in advance.

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# Homework Help: Another metrisation problem

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