Judyta Bąk

A metric space $X$ is {\em injective} if every non-expanding map $f \colon B\to X$ defined on a subspace $B$ of a metric space $A$ can be extended to a non-expanding map $\overline f \colon A\to X$. We prove that a metric space $X$ is a Lipschitz image of an injective metric space if and only if $X$ is {\em Lipschitz connected} in the sense that for every points $x,y\in X$, there exists a Lipschitz map $f \colon [0,1]\to X$ such that $f(0)=x$ and $f(1)=y$. In this case the metric space $X$ carries a well-defined intrinsic metric. A metric space $X$ is a Lipschitz image of a compact injective metric space if and only if $X$ is compact, Lipschitz connected and its intrinsic metric is totally bounded. A metric space $X$ is a Lipschitz image of a separable injective metric space if and only if $X$ is a Lipschitz image of the Urysohn universal metric space if and only if $X$ is analytic, Lipschitz connected and its intrinsic metric is separable.

We prove that a player $\alpha$ has a winning strategy in the Banach--Mazur game on a space $X$ if and only if $X$ is F-Y countably $\pi$-domain representable. We show that Choquet complete spaces are F-Y countably domain representable. We give an example of a space, which is F-Y countably domain representable, but it is not F-Y $\pi$-domain representable.