14 Sept 2004
2:30 - 4:00 M Barr
Characterizing Lindelof absolute CR-epics.
Abstract: There have been several talks during the past year or so by John Kennison and me on absolute CR-epic spaces, which are spaces X such that for any embedding X --> Y, the induced map C(Y) --> C(X) on the real-valued function rings is epic in the category CR of commutative rings. In his last talk, Kennison introduced a concept we called CNP (for countable neighbourhood property) that says that the intersection of any countable family of beta X-neighbourhoods of X is also a beta X-neighbourhood of X. Kennison showed that Lindelof CNP space is absolute CR-epic, but otherwise did not emphasize the property. Now, following an ingeneous suggestion of R. Levy we have shown that for Lindelof spaces, CNP is equivalent to absolute CR-epic. I will attempt to make this talk self-contained by proving both halves of the equivalence. It is worth mentioning that, while there are absolute CR-epic spaces that are not Lindelof, a necessary (but not sufficient) condition on such a space X is that be embeddable in a Lindelof absolute CR-epic space Y in such a way that C(X) = C(Y) and |Y-X| = 1.