Totally Bounded Quasi-Uniformities and Pseudocompactness of Bispaces |
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Authors: | Salvador Romaguera Manuel Sanchis |
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Institution: | (1) Universidad Politécnica de Valencia, Spain, ES;(2) Universidad Jaume I, Castellón, Spain, ES |
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Abstract: | We characterize pairwise Tychonoff bispaces that admit only totally bounded quasi-uniformities in terms of a suitable notion
of bitopological pseudocompactness. We also show that a pairwise Tychonoff bispace has a unique (up to equivalence) bicompactification
if and only if it admits a unique totally bounded quasi-unifomity. These results extend classical theorems of R. Doss for
uniform spaces to the quasi-uniform (bitopological) setting, and are applied to the study of T
0 topological spaces that admit a unique quasi-uniformity and a unique bicompactification, respectively. Finally, we discuss
the problem of extending the classical Glicksberg theorem on product of pseudocompact spaces to bispaces and a partial solution
is obtained.
Supported by the Spanish Ministry of Science and Technology, grant BFM2000-1111.
Supported by a grant from Generalitat Valenciana.
Received November 7, 2001; in revised form August 14, 2002 |
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Keywords: | 2000 Mathematics Subject Classification: 54E55 54E15 54D35 54B10 |
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