From: Allan Adler <ara@nestle.ai.mit.edu>
Subject: Re: examples of totally disconnected compact Hausdorff spaces
Date: 08 Feb 2000 01:35:58 -0500
Newsgroups: sci.math
Summary: [missing]

Stephen Paul King <stephenk1@home.com> writes:

>       Could someone give me a few examples of totally disconnected
> compact Hausdorff spaces?

(0) A finite discrete set with (at least) 2 elements.
(1) The infinite sequence 1,1/2,1/3,1/4,1/5,... together with its
    limit 0, with the relative topology from the reals.
(2) The 1 point compactification of an infinite discrete space.
    Example (1) arises in this way.
(3) The cantor set, in various guises (e.g. the p-adic numbers).
(4) The Stone-Cech compactification of an infinite discrete space.
    This is an example of an "extremely disconnected space": the closure
    of every open set is open.
(5) The group of automorphisms of the field of all algebraic
    numbers, with the Krull topology. This is one of the guises of (3).
(6) Any closed subspace of a totally disconnected compact Hausdorff space.
(7) Any product of totally disconnected compact Hausdorff spaces.

Combining (0), (6) and (7) leads to:
(8) The set of all maximal ideals of a Boolean algebra.

Every totally disconnected compact Hausdorff space arises as an example
of (8), up to homeomorphism, according to the Stone Representation Theorem
for Boolean algebras. Given a totally disconnected compact Hausdorff
space X, one can recover the boolean algebra as the boolean agebra of
simultaneously open and closed subsets of X. This is explained in
Halmos' nice book Lectures on Boolean Algebras, which is where I
learned most of what I know about it. He also shows how to give
other exotic examples, such as totally disconnected compact Hausdorff
spaces which have no autohomeomorphisms other than the identity map.

Allan Adler
ara@zurich.ai.mit.edu

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