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C * -points vs P -points and P ♭ -points

Jorge Martinez, Warren Wm. McGovern (2022)

Commentationes Mathematicae Universitatis Carolinae

In a Tychonoff space X , the point p ∈ X is called a C * -point if every real-valued continuous function on C ∖ { p } can be extended continuously to p . Every point in an extremally disconnected space is a C * -point. A classic example is the space 𝐖 * = ω 1 + 1 consisting of the countable ordinals together with ω 1 . The point ω 1 is known to be a C * -point as well as a P -point. We supply a characterization of C * -points in totally ordered spaces. The remainder of our time is aimed at studying when a point in a product space is a C * -point....

Characterizations of Some Classes of Perfect Spaces in Terms of Continuous Selections Avoiding Supporting Sets

Takamitsu Yamauchi (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

Some kinds of perfect spaces, including paracompact perfectly normal spaces and collectionwise normal perfect spaces, are characterized in terms of continuous selections avoiding supporting sets. A necessary and sufficient condition on a domain space for a selection theorem of E. Michael [Fund. Math. 47 (1959), 173-178] to hold is also obtained.

Closed graph multi-selections

Valentin Gutev (2011)

Fundamenta Mathematicae

A classical Lefschetz result about point-finite open covers of normal spaces is generalised by showing that every lower semi-continuous mapping from a normal space into the nonempty compact subsets of a metrizable space admits a closed graph multi-selection. Several applications are given as well.

Completely regular spaces

H. L. Bentley, Eva Lowen-Colebunders (1991)

Commentationes Mathematicae Universitatis Carolinae

We conduct an investigation of the relationships which exist between various generalizations of complete regularity in the setting of merotopic spaces, with particular attention to filter spaces such as Cauchy spaces and convergence spaces. Our primary contribution consists in the presentation of several counterexamples establishing the divergence of various such generalizations of complete regularity. We give examples of: (1) a contigual zero space which is not weakly regular and is not a Cauchy...

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