October 16th, 2000

beartato phd


Aha, so a space which has a countable basis and which can
simultaneously separate any countable collection of isolated
points must be regular. You look at any closed C and point x
which by hypothesis can't be separated, choose a chain of
neighborhoods U_i of x such that one of them fits inside any
neighborhood of x, realize that they must meet successive
V_i each owning a c_i of C, and then observe by sigma-s.s.
that we must be able to find a c_j in the closure of infintely
many other c_i (i ne j) so it has to be in the closure of
infinitely (and cofinally!) many U_i, violating Hausdorff, QED.