From: <kp@pascal.math.yorku.ca>
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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