From:
Subject: Re: Totally disconnected spaces
Date: Tue, 1 Jun 1999 09:16:30 -0400
Newsgroups: sci.math.research
Keywords: zero-dimensional spaces
On 27 May 1999, Andrej Bauer wrote:
>
> I have encountered various questions about totally disconnected
> topological spaces. Any classical or well-known texts that deal
> with totally disconnected spaces in more than just a passing way,
> and in sufficient generality (see below), would be very welcome.
Engelking's `General Topology' contains a comprehensive discussion
of `various kinds of disconnectedness' in Chapter 6.
>
> Definition 1:
> A space X is *totally disconnected* if for every two points x and y
> there exists a clopen (closed and open) set U that contains x and does
> not contain y.
This is the commonly accepted definition of total disconnectivity.
>
> Definition 2:
> A space X is *totally disconnected* if it has a topological basis
> consisting of clopen sets.
This notion is commonly known as zero-dimensionality.
>
> Question 1: Is it true in general that Def. 1 implies Def. 2?
>
No, a famous example is the set of points in $\ell_2$ all of whose
coordinates are rational --- this is due to Erd\H{o}s.
Cheers,
KP
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