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

A non-negative function d:X²→ R d : X 2 → R is called a hemimetric if $$d(x,x)=0$$ d ( x , x ) = 0 and d(x,y)≤ d(x,z)+d(z,y) d ( x , y ) ≤ d ( x , z ) + d ( z , y ) for any x,y,z∈ 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 _∞ ^_⊕(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∈ Y b ∈ Y , Z is a regular hemimetrizable space and f:X× Y→ 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× \ 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 β β -unfavorable for the Christensen σ σ -game space. More precisely, if X is a β β -unfavorable space for the Christensen σ σ -game and f:X→ Y f : X → Y continuous with respect to a hemimetric d on Y , then there is a dense G_δ 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 d=max ,d'\ d ^ = max { d , d ′ } . Moreover, we show that this result cannot be extended to an α α -favorable space for the Saint-Raymond game.