Subject: Hovey's first question
Date: Fri, 10 Oct 2003 08:34:37 -0400
From: Tom Goodwillie
To: Don Davis
If X is completely regular (for example, CW), then the Stone-Cech
compactification BX is a compact Hausdorff space that contains X as a
dense subset. It has the marvelous property that it is the universal
example of a compact Hausdorff space with a map from X.
It turns out that there can never be a path in BX that begins in X
and ends outside X, and therefore if X is path-connected then X is a
path-component of BX.
This you-can't-get-there-from-here feature follows from the fact that
a sequence in X can never converge in BX to a point outside X. That
in turn follows from the statement that if a sequence in X does not
converge (in X) then there are two disjoint closed sets each
containing a subsequence.
So here's two more questions:
1. What groups can be pi_1 of a path-connected compact Hausdorff space?
2. What weak homotopy types are represented by compact Hausdorff spaces?
- Tom Goodwillie