Products and Sums of Ultracomplete Topological Spaces
by
David Buhagiar
Mathematics Department, Okayama University, Okayama 700-8530
Coauthors: Iwao Yoshioka (Mathematics Department, Okayama University, Okayama 700-8530)

In 1995 S.Romaguera introduced the notion of cofinally Cech complete topological spaces and he showed that a metrizable space admits a cofinally complete metric (otherwise called ultracomplete metric), a term introduced independently by N.R.Howes in 1971 and A.Császár in 1975, if and only if it is cofinally Cech complete. In a recent paper by the authors (1999), the notion of ultracomplete topological space was introduced as Tychonoff spaces which have countable character in any one (equivalently, in all) of their compactifications. It was proved in the same paper that the class of ultracomplete topological spaces coincide with the class of cofinally Cech complete topological spaces. This external characterization of cofinally Cech complete topological spaces shows clearly that the class of cofinally Cech complete spaces lies between the class of locally compact spaces and the class of Cech complete spaces. It was shown that in the realm of Tychonoff spaces, cofinally Cech complete spaces (here called ultracomplete spaces) are invariant under open maps and are invariant/inverse invariant under perfect maps. It was also shown that like Cech complete spaces, ultracomplete spaces are not invariant under closed maps. In this paper we study both products and sums of ultracomplete topological spaces. We show that ultracomplete spaces do not behave well under Tychonoff products, in fact the product of two ultracomplete spaces is not necessarily ultracomplete. It is proved that if X and Y are ultracomplete, non-locally compact spaces, then a necessary and sufficient condition for the Tychonoff product X×Y to be ultracomplete is that both X and Y are countably compact. Other general results concerning products and sums are given together with examples to clarify these results.

Date received: June 30, 1999

Copyright © 1999 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 # caby-30.