Categorically related topologies and hemimetrical analogues of the Baire and Kenderov theorems

Abstract

AbstractA non-negative function $$d:X^2\rightarrow {\mathbb {R}}$$ d : X 2 → R is called a hemimetric if $$d(x,x)=0$$ d ( x , x ) = 0 and $$d(x,y)\le d(x,z)+d(z,y)$$ d ( x , y ) ≤ d ( x , z ) + d ( z , y ) for any $$x,y,z\in X$$ x , y , z ∈ X . Then $$d'(x,y)=d(y,x)$$ d ′ ( x , y ) = d ( y , x ) is a hemimetric as well. We call it the dual hemimetric. We show the existence of the universal hemimetric space $$\ell _\infty ^{^{_\oplus }}(T)$$ ℓ ∞ ⊕ ( T ) which allows to prove the hemimetrizability of a premetric space with the weak triangle inequality. Then we generalize the Baire horizontal theorem and obtain some version of the Baire curves theorem for a separately continuous function ranged in a regular hemimetrizable space. In particular, we show that if X is a topological space, Y is a first countable space at $$b\in Y$$ b ∈ Y , Z is a regular hemimetrizable space and $$f:X\times Y\rightarrow Z$$ f : X × Y → Z is a separately continuous function, then there is a residual set A in X such that f is continuous at every point of $$A\times \{b\}$$ A × { b } . We also generalize the Kenderov theorem on the Čech complete spaces and prove that that the open ball topology for the dual hemimetric $$d'$$ d ′ is categorically related to the open ball topology for a hemimetric d with respect to any $$\beta $$ β -unfavorable for the Christensen $$\sigma $$ σ -game space. More precisely, if X is a $$\beta $$ β -unfavorable space for the Christensen $$\sigma $$ σ -game and $$f:X\rightarrow Y$$ f : X → Y continuous with respect to a hemimetric d on Y, then there is a dense $$G_\delta $$ G δ -set A in X such that f is continuous on A with respect to the dual hemimetric $$d'$$ d ′ at every point of A. Therefore, f is continuous on A with respect to the pseudometric $${\widehat{d}}=\max \{d,d'\}$$ d ^ = max { d , d ′ } . Moreover, we show that this result cannot be extended to an $$\alpha $$ α -favorable space for the Saint-Raymond game.