Lindelöf \Sigma-property in C_p(X) and countable spread of X
by
Vladimir Tkachuk
Universidad Autonoma Metropolitana de Mexico

All spaces are assumed to be Tychonoff. A space X is Lindelöf \Sigma if it is a continuous image of a space which can be mapped perfectly onto a second countable one. The spread s(X) of a space X is the supremum of cardinalities of discrete subspaces of X. In [Ar] it is proved that if C_p(X) is a Lindelöf \Sigma-space and the spread of C_p(X) is countable, then X is has a countable network. The condition s(C_p(X))=\omega implies s(X²)=\omega so Arhangelskii asks [Ar, Problem 11] whether X has a countable network in case s(X)=\omega and C_p(X) is a Lindelöf \Sigma-space. He proved that the answer is ÿes" under MA+ not CH or in ZFC if s(X×X)=\omega or X is a topological group.

We prove in ZFC that if the spread of a space X is countable and C_p(X) is a Lindelöf \Sigma-space then X is has a countable network. This gives a complete answer to Problem 11 from [Ar].

Reference

[Ar] A.V. Arhangel'skii, On Lindelöf property and spread in C_p-theory, Topology Appl., 74:1(1996), 83-90.

Date received: March 23, 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 # cagw-04.