Pugh's definition of 'totally disconnected space'

Question

In Pugh's text Real Mathematical Analysis page 105, he defines a 'totally disconnected space' as follows:

A metric space $M$ is totally disconnected if each point $p ∈ M$ has arbitrarily small clopen neighborhoods. That is, given $\epsilon > 0, p ∈ M$, there exists a clopen set $U$ such that $p ∈ U ⊂ M_{\epsilon}(p)$

Edit: $M_{\epsilon}(p)$ means 'an open ball of radius $\epsilon$ around p'

The usual definition of a totally disconnected space is one where the singletons are the only connected subspaces.

I can see how Pugh's definition implies the usual one, but not the other way around. There may be a contrived example where in fact the clopen subsets are ugly enough that they allow for total disconnectedness yet there aren't any clopen sets contained in a ball of radius $\epsilon$.

I can't find a counterexample either since all the metric spaces I've worked with thus far are 'nice'.

So my question is - with regards to metric spaces, are these definitions equivalent?

https://math.stackexchange.com/questions/142070/totally-disconnected-implies-base-of-closed-sets/151954#151954 — Moishe Kohan, Jan 16 '20 at 16:31

score 5 · Accepted Answer · answered Jan 16 '20 at 16:27

5

Pugh's notion is normally called $0$-dimensional (clopen base) and totally disconnected is usually reserved for spaces whose components are singletons.

A (separable metric) space where these notions differ is Erdős space, which is not zero-dimensional (but $1$-dimensional) but totally disconnected. There are even infinite-dimensional Polish spaces that are totally disconnected (!). In locally compact spaces the notions do coincide, but too bad that Pugh uses non-standard terminology.

answered Jan 16 '20 at 16:27

Henno Brandsma

242,131

could you provide a hint about how to prove that Pugh's definition implies the usual definition? Thanks! – Karthik Kannan Apr 15 '21 at 10:29
@KarthikKannan if a component would have two distinct points there’d be a clopen set separating making the component disconnected (contradiction). – Henno Brandsma Apr 15 '21 at 11:06
I am extremely sorry, I meant the other way around i.e. usual definition implies Pugh's definition. Like the OP I couldn't figure the proof of this direction. Thank you. – Karthik Kannan Apr 15 '21 at 11:20
@KarthikKannan I gave a metric counterexample, so you cannot show the other way around because it's false. – Henno Brandsma Apr 15 '21 at 11:21
Thank you very much. I understood your point. – Karthik Kannan Apr 15 '21 at 11:25

Pugh's definition of 'totally disconnected space'

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