Continuous Selections and Extremally Disconnected Spaces

Adolfo Pimienta () and Manuel Sanchis
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Adolfo Pimienta: Facultad de Ciencias Básicas y Biomédicas, Universidad Simón Bolivar, Barranquilla 080002, Colombia
Manuel Sanchis: Institut de Matemàtiques i Aplicacions de Castelló (IMAC), Universitat Jaume I, 12071 Castelló, Spain

Mathematics, 2023, vol. 11, issue 4, 1-8

Abstract: 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 .

Keywords: selection theory; extremally disconnected spaces; extremally disconnected P-spaces (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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