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