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Extension of multisequences and countably uniradial classes of topologies

Szymon Dolecki;  Andrzej Starosolski;  Stephen W. Watson — 2003

Commentationes Mathematicae Universitatis Carolinae

It is proved that every non trivial continuous map between the sets of extremal elements of monotone sequential cascades can be continuously extended to some subcascades. This implies a result of Franklin and Rajagopalan that an Arens space cannot be continuously non trivially mapped to an Arens space of higher rank. As an application, it is proved that if for a filter ℋ on ω , the class of ℋ -radial topologies contains each sequential topology, then it includes the class of subsequential topologies....

A metrizable completely regular ordered space

Hans-Peter A. Künzi;  Stephen W. Watson — 1994

Commentationes Mathematicae Universitatis Carolinae

We construct a completely regular ordered space ( X , 𝒯 , ≤ ) such that X is an I -space, the topology 𝒯 of X is metrizable and the bitopological space ( X , 𝒯 ♯ , 𝒯 ♭ ) is pairwise regular, but not pairwise completely regular. (Here 𝒯 ♯ denotes the upper topology and 𝒯 ♭ the lower topology of X .)

Some remarks on the product of two C α -compact subsets

Salvador García-Ferreira;  Manuel Sanchis;  Stephen W. Watson — 2000

Czechoslovak Mathematical Journal

For a cardinal α , we say that a subset B of a space X is C α -compact in X if for every continuous function f X → ℝ α , f [ B ] is a compact subset of ℝ α . If B is a C -compact subset of a space X , then ρ ( B , X ) denotes the degree of C α -compactness of B in X . A space X is called α -pseudocompact if X is C α -compact into itself. For each cardinal α , we give an example of an α -pseudocompact space X such that X × X is not pseudocompact: this answers a question posed by T. Retta in “Some cardinal generalizations of pseudocompactness”...

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