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Continuous Selections and Extremally Disconnected Spaces

Author

Listed:
  • 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)

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 .

Suggested Citation

  • Adolfo Pimienta & Manuel Sanchis, 2023. "Continuous Selections and Extremally Disconnected Spaces," Mathematics, MDPI, vol. 11(4), pages 1-8, February.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:4:p:791-:d:1057607
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    References listed on IDEAS

    as
    1. M. Ali Khan & Metin Uyanik, 2021. "The Yannelis–Prabhakar theorem on upper semi-continuous selections in paracompact spaces: extensions and applications," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 71(3), pages 799-840, April.
    2. Luisa Di Piazza & Kazimierz Musiał, 2020. "Decompositions of Weakly Compact Valued Integrable Multifunctions," Mathematics, MDPI, vol. 8(6), pages 1-13, May.
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