On closures of discrete subsets
by
Ofelia T. Alas
Universidade de São Paulo

In O.T.Alas, V.V.Tkachuk and R.G.Wilson, Closure of discrete sets often reflect global properties (to appear in Top.Proc.), we look for topological properties P such that if X is a Hausdorff space and the closure of every discrete subset has property P, then X also has property P. Here some topological cardinal inequalities are established which imply partial results on the precedent type of problem. Let X be a Hausdorff space and let L, t, F and s denote the Lindelöf degree, tightness, lenght of free sequences and spread of X, respectively. Define LF(X) as the supremum of { L(cl D) : D discrete in X } and tF(X) as the supremum of { t(cl D) : D discrete in X } .

Theorem. The following inequalities hold:

1) L(X) <= LF(X) F(X);

2) F(X) <= LF(X) tF(X).

Corollary. If X has countable tightness and for every discrete D, cl D is Lindelöf, then X is also Lindelöf.

Theorem. If X is a weakly discretely generated Lindelöf space, c is a regular cardinal, and  if    ^ s(X) <= c, then the density of X is not bigger than c.

Corollary. (MA + not CH) If X is compact, ^ s(X) <= c and the cardinality of the closure of every discrete subset is not bigger than c, then the cardinality of X is not bigger than c.

Other results are proven.

Date received: February 22, 2001

Copyright © 2001 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 # cafw-23.