Discrete subspaces and topologies they determine
by
Vladimir Tkachuk
Universidad Autonoma Metropolitana de Mexico
Coauthors: A.Dow, M.G. Tkachenko, R.G. Wilson

The results presented in this talk are obtained jointly with A. Dow, M.G. Tkachenko and R.G. Wilson. It is natural to say that the topology of a space X is determined by discrete subspaces if for every A subset X the closure of A is the union of the closures of discrete subspaces of A. We will also call such spaces discretely generated. There are two important classes of discretely generated topological spaces: Fréchet-Urysohn spaces and the scattered ones. In a Fréchet-Urysohn space X every point x from a closure of a set A subset X is the limit of a convergent sequence S subset A. Clearly, S is a discrete subspace of X. If X is scattered then every subspace of X has a dense discrete subspace and so every point of [`A] is in the closure of a discrete subspace of A. The purpose of this talk is to present some results about the classes of discretely generated and weakly discretely generated spaces which are both wider than either the class of Fréchet-Urysohn spaces or the class of scattered ones.

We prove that the (weak) discrete generability is (closed) hereditary; every compact space is weakly discretely generated and not necessarily discretely generated. Another result is that every compact space of countable tightness is discretely generated. We also show that under the Continuum Hypothesis any dyadic discretely generated compact space is metrizable.

We give examples which show that a pseudocompact Tychonoff space can fail to be weakly discretely generated. We show that the same can happen to a countably compact Hausdorff space. We prove that it is consistent with ZFC that there are countably compact Tychonoff spaces which are not weakly discretely generated. This result is related to the theory of remote points, because it is easy to see that the existence of countably compact Tychonoff non-weakly discretely generated spaces is equivalent to the existence of a countably compact space X with ``discretely remote points" in \betaX, i.e., the points of \betaX\X which are not in the closure of any discrete subspace of X.

Date received: June 3, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caeu-13.