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?
Best Answer
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.