Reflecting global properties by closures of discrete subsets
by
Vladimir Tkachuk
Universidad Autónoma Metropolitana de Mexico
Coauthors: Ofelia T. Alas (University of São Paulo, São Paulo, Brazil), Richard G. Wilson (Universidad Autónoma Metropolitana,Unidad Iztapalapa, México D.F.)

We call a topological property P, discretely reflexive in a class A, if, for any X in A, the space X has P if and only if the closures of all discrete subspaces of X have P. It is know that compactness is discretely reflexive in the class of Hausdorff spaces while it is an open problem (posed by Arhangel'skii) whether the Lindelöf property is discretely reflexive in the class of normal spaces.

We prove, in particular, that

1) the hereditary Lindelöf degree, the initial \kappa-compactness and sequential compactness are discretely reflexive in the class of Hausdorff spaces;

2) tightness, sequentiality, character and the Fréchet-Urysohn property are discretely reflexive in the class of compact Hausdorff spaces;

3) it is independent of ZFC whether metrizability is discretely reflexive in the class of compact Hausdorff spaces.

We also give examples of various properties which are not discretely reflexive and give some applications of discrete generability.

Date received: July 21, 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-65.