arXiv subjects

Judyta Bąk

Publications and source records attributed to Judyta Bąk.

4 recordsLinked to original sources

arXiv2026

Topological size of the set of universal and ultrahomogeneous retractions on the Urysohn space

In this paper, we investigate the set $\mathcal{U}(\mathbb{U})$ of universal and ultrahomogeneous $1$-Lipschitz retractions acting on the Urysohn space as the subspace of the space $\mathcal{R}(\mathbb{U})$ of all $1-$Lipschitz retractions defined on the Urysohn space. Especially, we study Borel complexity and density $\mathcal{U}(\mathbb{U})$ in $\mathcal{R}(\mathbb{U}).$ In order to do that, we introduce a new extension property $(UR^*)$ that is equivalent to the universality and ultrahomogeneity of a retraction, and a new pointwise retract topology.

math.GN↗

arXiv2024

Characterizing Lipschitz images of injective metric spaces

A metric space $X$ is {\em injective} if every non-expanding map $f:B\to X$ defined on a subspace $B$ of a metric space $A$ can be extended to a non-expanding map $\bar f:A\to X$. We prove that a metric space $X$ is a Lipschitz image of an injective metric space if and only if $X$ is Lipschitz connected in the sense that for every points $x,y\in X$, there exists a Lipschitz map $f:[0,1]\to X$ 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.

math.GN↗

arXiv2019

Topological spaces with the Freese--Nation property II

The aim of this paper is to study the class of spaces with the FNS property and $π-\FNS$ property. We shown that compact spaces with the FNS property for some base consisting of cozero-sets are openly generated spaces and spaces with the $π-\FNS$ property are skeletally generated spaces.

math.GN↗

arXiv2018

The Banach--Mazur game and the strong Choquet game in domain theory

We prove that a player $α$ has a winning strategy in the Banach--Mazur game on a space $X$ if and only if $X$ is F-Y countably $π$-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 $π$-domain representable.

math.GN↗