Categorically related topologies and hemimetrical analogues of the Baire and Kenderov theorems
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}, PLISKA, Stud. Math. Bulg. 1, 122--127 (1977; Zbl 0493.41044)]. Assume that \(X\) is a \(\beta\)-unfavorable space for the Christensen \(s\)-game, \((Y,d)\) is a hemimetric space, \(d'(x,y)=d(x,y)\) for \(x,y\in Y\), and \(f\colon X\to Y\) is a \(\tau_d\)-continuous function. Then there exists a dense \(G_\delta\) subset \(A\subset X\) such that \(f\) is \(\tau_{d'}\)-continuous at every point of \(A\).\N\NIn the last part, the author studies analogous problems for multi-valued maps. The paper ends with a list of six open problems.
scientific article; zbMATH DE number 7887434
scientific article