Weak selections and suborderable metrizable spaces

Gutev, Valentin

doi:10.1016/j.topol.2020.107506

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2023

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“…Its applications range from Functional Analysis to the theory of discrete dynamical systems, passing through the fixed point theory. Among others, the interested reader is referred to [1][2][3][4][5][6][7][8]. In this paper, we deal with selection theory, as introduced by Michael in his classic paper [9]: related to one of the most important subjects in topology, the problem of extension of continuous functions.…”

Section: Introductionmentioning

confidence: 99%

Continuous Selections and Extremally Disconnected Spaces

Pimienta

Sanchis

2023

Mathematics

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This paper deals with extremally disconnected spaces and extremally disconnected P-spaces. A space X is said to be extremally disconnected if, for every open subset V of X, the closure of V in X is also an open set. P-spaces are spaces in which the intersection of countably many open sets is an open set. The authors present a new characterization of extremally disconnected spaces, and the extremally disconnected P-spaces, by means of selection theory. If X is either an extremally disconnected space or an extremally disconnected P-space, then the usual theorems of extension of real-valued continuous functions for a dense subset S of X can be deduced from our results. A corollary of our outcomes is that every nondiscrete space X of nonmeasurable cardinality has a dense subset S such that S is not C-embedded in X.

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