Pointfree pseudocompactness revisited
- Authors: Dube, T , Matutu, Phethiwe P
- Date: 2009
- Language: English
- Type: Article
- Identifier: vital:6781 , http://hdl.handle.net/10962/d1006924
- Description: We give several internal and external characterizations of pseudocompactness in frames which extend (and transcend) analogous characterizations in topological spaces. In the case of internal characterizations we do not make reference (explicitly or implicitly) to the reals.
- Full Text:
- Date Issued: 2009
- Authors: Dube, T , Matutu, Phethiwe P
- Date: 2009
- Language: English
- Type: Article
- Identifier: vital:6781 , http://hdl.handle.net/10962/d1006924
- Description: We give several internal and external characterizations of pseudocompactness in frames which extend (and transcend) analogous characterizations in topological spaces. In the case of internal characterizations we do not make reference (explicitly or implicitly) to the reals.
- Full Text:
- Date Issued: 2009
A few points on Pointfree pseudocompactness
- Authors: Dube, T , Matutu, Phethiwe P
- Date: 2007
- Language: English
- Type: Article
- Identifier: vital:6783 , http://hdl.handle.net/10962/d1006926
- Description: 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.
- Full Text:
- Date Issued: 2007
- Authors: Dube, T , Matutu, Phethiwe P
- Date: 2007
- Language: English
- Type: Article
- Identifier: vital:6783 , http://hdl.handle.net/10962/d1006926
- Description: 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.
- Full Text:
- Date Issued: 2007
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