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