Categorically related topologies and hemimetrical analogues of the Baire and Kenderov theorems
type: Item
instance of: scholarly article
title: Categorically related topologies and hemimetrical analogues of the Baire and Kenderov theorems
zbMATH Open document ID: 1548.54015
author: O. V. Maslyuchenko
published in: Results in Mathematics
publication date: 2024-07-25T00:00:00Z
review text: The reviewed article is devoted to generalizations of the famous Baire theorem on the sets of continuity points of separately continuous functions. The author studies here separately continuous functions with values in hemimetric spaces, see [\textit{T. Richmond}, General topology. An introduction. Berlin: De Gruyter (2020; Zbl 1454.54001)]. Recall that a function \(d\colon X\times X\to\mathbb{R}_+\) is called a \textit{hemimetric} if it satisfies two conditions: \(d(x,x)=0\) for every \(x\in X\) and \(d(x,y)\le d(x,z)+d(z,y)\) for any \(x,y,z\in X\). A topological space is \textit{hemimetrizable} if the topology of \(X\) coincides with the open ball topology for some hemimetric.\N\NThe paper can be divided into three parts. In the beginning sections the author gives some definitions, facts and examples concerning general theory of hemimetric spaces. The main result here says that for any \(T\neq\emptyset\) the space \(\ell_\infty(T)\) is isometrically universal for the class of all \(T_0\) hemimetric spaces of cardinality \(\le |T|\).\N\NThen the author proves some facts concerning the sets of continuity points of separately continuous functions ranged in hemimetric spaces, analogous to classical Baire theorems. For example, the following theorem is proved. Assume that \(X\) is a topological space, \(Y\) is metrizable, \(Z\) is a regular (\(T_3\)) hemimetrizable space, \(g\colon X\to Y\) is continuous, and \(f\colon X\times Y\to Z\) is separately continuous and such that the map \(x\mapsto f(x,g(x))\) is continuous. Then the set \[\{ x\in X \colon f \text{ is continuous at }(x,g(x))\}\] is residual in \(X\).\N\NNext, the author considers different versions of Choquet, Christensen and Saint-Raymond games, see [\textit{J. P. R. Christensen}, Proc. Am. Math. Soc. 82, 455--461 (1981; Zbl 0472.54007)], and [\textit{J. Saint-Raymond}, ibid. 87, 499--504 (1983; Zbl 0511.54007)], and proves the following generalization of Kanderov's theorem, see [\textit{P. S. Kenderov}, P…
reviewed by: Tomasz Natkaniec
Mathematics Subject Classification ID: 54C20
Mathematics Subject Classification ID: 54C10
Mathematics Subject Classification ID: 54C30
Mathematics Subject Classification ID: 26A15
zbMATH DE Number: 7887434
zbMATH Keywords: separately continuous function
zbMATH Keywords: topological games
zbMATH Keywords: upper semicontinuous
zbMATH Keywords: lower semicontinuous
zbMATH Keywords: quasimetric
zbMATH Keywords: hemimetric
zbMATH Keywords: Christensen game
zbMATH Keywords: categorically related topologies
zbMATH Keywords: hemimetrizable space
zbMATH Keywords: premetric
zbMATH Keywords: Saint-Raymond game
MaRDI profile type: Publication
cites work: On functions of real variables.
cites work: Q3854306
cites work: On existence of separately continuous mappings at curves
cites work: Category theorems
cites work: Q5806961
cites work: Separate continuity and joint continuity
cites work: Joint Continuity of Separately Continuous Functions
cites work: Jeux Topologiques et Espaces de Namioka
cites work: Espaces de Baire et espaces de Namioka
cites work: Points de Continuite d'Une Fonction Separement Continue
cites work: Points de Continuite d'Une Fonction Separement Continue. II
cites work: Q3479846
cites work: Q3488132
cites work: Q4289125
cites work: Notes sur la Propriete de Namioka
cites work: Joint continuity of \(KC\)-functions
cites work: Upper Namioka property of compact-valued mappings
cites work: Namioka spaces, \textit{GO}-spaces and an \(o\)-game
cites work: Q3955829
cites work: MULTI-VALUED MAPPINGS AND PROPERTIES OF THEM SIMILAR TO CONTINUITY
cites work: Sur quelques points du calcul fonctionnel.
cites work: On the ``third axiom of metric space''.
cites work: Über die Axiome des metrischen Raumes.
cites work: On quasi-metric spaces.
cites work: On semi-metric spaces.
cites work: Der allgemeine Metrisationssatz und das Symmetrieaxiom.
cites work: MAPPINGS AND SPACES
cites work: Q5532985
cites work: Q5610290
cites work: Q5656091
cites work: Q5660870
cites work: The geodesic problem in quasimetric spaces
cites work: A new fixed point theorem on generalized quasimetric spaces
cites work: Non-Hausdorff Topology and Domain Theory
cites work: General Topology
cites work: Discontinuity points of separately continuous mappings with values in the Sorgenfrey line
cites work: Q3994935
DOI: 10.1007/S00025-024-02197-1
description: scientific article; zbMATH DE number 7887434