A few points on Pointfree pseudocompactness

Dube, T. and Matutu, P. (2007) A few points on Pointfree pseudocompactness. Quaestiones Mathematica, 30 (4). pp. 451-464. ISSN 1607-3606

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Abstract

We present several characterizations of completely regular pseudocompact frames. The first is an extension to frames of characterizations of completely regular pseudocompact spaces given by Väänänen. We follow with an embedding-type characterization stating that a completely regular frame is pseudocompact if and only if it is a P-quotient of its Stone-Čech compactification. We then give a characterization in terms of ideals in the cozero parts of the frames concerned. This characterization seems to be new and its spatial counterpart does not seem to have been observed before. We also define relatively pseudocompact quotients, and show that a necessary and sufficient condition for a completely regular frame to be pseudocompact is that it be relatively pseudocompact in its Hewitt realcompactification. Consequently a proof of a result of Banaschewski and Gilmour that a completely regular frame is pseudocompact if and only if its Hewitt realcompactification is compact, is presented without the invocation of the Boolean Ultrafilter Theorem.

Item Type:Article
Additional Information:The second-named author expresses her gratitude for financial assistance from Rhodes University and the National Research Foundation of South Africa.
Uncontrolled Keywords:frame, pseudocompact, relatively pseudocompact, P-quotient, ideal, maximal ideal.
Subjects:Y Unknown > Subjects to be assigned
Divisions:Faculty > Faculty of Science > Mathematics (Pure & Applied)
ID Code:842
Deposited By: Mrs Eileen Shepherd
Deposited On:11 Jan 2008
Last Modified:06 Jan 2012 16:19
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