Subject: Stone-Cech compactifications
Date: Sun, 12 Oct 2003 22:46:59 -0400
From: Tom Goodwillie
To: Don Davis
Hi everybody. I've embarrassed myself again by shooting my mouth off.
I was so happy to have learned a few cool facts about point set
topology that I went and overstated some things. I shouldn't have
been so hasty to show off my new-found knowledge.
I made two mistakes. First, to argue that sequences cannot escape
from X in BX I should have assumed X was normal. Second, and more
importantly, it's not clear that paths cannot escape just because
sequences cannot escape.
Let me share a few things that I have learned. Probably everybody
should know most of this this stuff, but I didn't, and I know I'm not
the only topologist who knows very little about general topology, so
I figure somebody might get something out of this.
1. For any space X there is a space called BX (well, it should be
beta but B is easier to type), defined as a subspace of the product
of a lot of copies of I, one for each continuous map from X to I. BX
is the closure of the image of the obvious map from X to this product.
2. BX is compact Hausdorff, of course, and B is a functor from spaces
to compact Hausdorff spaces. It is left adjoint to the forgetful (or
inclusion) functor.
3. The canonical map from X to BX is an embedding (i.e. a
homeomorphism followed by the inclusion of a subspace) if and only if
X is completely regular (i.e. if in X points are closed and points
can be separated from closed sets by real functions).
In this case (at least) BX is called the Stone-Cech compactification of X.
4. It follows from 3 that a implies d below, and therefore a, b, c, d
are equivalent
a. X is completely regular
b. X is a subspace of a completely regular space
c. X is a subspace of a normal space
d. X is a subspace of a compact Hausdorff space
Note that regularity and complete regularity are (obviously)
inherited by subspaces, but normality is not.
5. If X is regular then for any non-convergent sequence in X there
are disjoint closed sets A and B in X each of which contains some
subsequence. (Case 1: The sequence has a limit point x. Then since
the sequence does not converge to x, there is an open nbhd X-A of X
whose complement A contains a subsequence. Since X is regular, X-A
contains a closed nbhd B of X. Since x is a limit point, B contains a
subsequence. Case 2: There is no limit point. Then the set of points
in the sequence is closed, discrete, and infinite, so write as the
union of two infinte sets A and B ...)
6. If X is normal then, using a function f:X->I that equals 0 on A
and 1 on B, we conclude from 5 that a sequence in X cannot converge
in BX except by converging in X.
7. I wanted to conclude from 6 that a path in BX cannot be partly in
X and partly out of X. But I was foolishly imagining that X was open
in BX. Silly me! That would make X an open subset of a compact
Hausdorff space, therefore locally compact. I don't know why you
can't have a path p:I->BX such that p(t) is only in X when t is 0. On
the other hand, I can't think of an example.
8. The colleague of Rudyak who first brought this up says that he can
rule such things out in the case when X is paracompact. So I gather
that his assertion is, as I thought, that a connected CW complex is a
path component of its Stone-Cech compactification.
Tom Goodwillie