When is |C(X ×Y)| = |C(X)| ×|C(Y)| ?
by
W.W. Comfort
Wesleyan University
Coauthors: Salvador García-Ferreira, Melvin Henriksen, Richard G. Wilson, R. Grant Woods

[We report on work in progress.]

Positive Results.

Theorem. For Tychonoff spaces X and Y, the relation |C(X ×Y)| = |C(X)| ·|C(Y)| holds if either X or Y is separable, or if X ×Y is either pseudocompact, metrizable, or weakly Lindelof.

Negative Results.

Definition. An ordered pair (m, t) of cardinal numbers is a bad cardinal pair if m^t > m^\omega = m > 2^t.

Theorem.

1. For every cardinal \kappa there is a bad cardinal pair (m, t) such that t > \kappa.
2. For every bad cardinal pair (m, t) there are locally compact Tychonoff spaces X and Y such that |C(X ×Y)| = m^t > mômega = m = |C(X)| > |C(Y)| = 2^t; and
3. For m, t, X and Y as in (1), the "disjoint union" space Z : = X + Y satisfies |C(Z ×Z)| = m^t > m = |C(Z)|.

Date received: March 4, 1997

Copyright © 1997 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 # caak-55.