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Joanna Garbulińska-Węgrzyn

Publications and source records attributed to Joanna Garbulińska-Węgrzyn.

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

A universal Banach space with a $K$-unconditional basis

For a constant $K\geq 1$ let $\mathfrak{B}_K$ be the class of pairs $(X,(\mathbf e_n)_{n\inω})$ consisting of a Banach space $X$ and an unconditional Schauder basis $(\mathbf e_n)_{n\inω}$ for $X$, having the unconditional basic constant $K_u\leq K$. Such pairs are called $K$-based Banach spaces. A based Banach space $X$ is rational if the unit ball of any finite-dimensional subspace spanned by finitely many basic vectors is a polyhedron whose vertices have rational coordinates in the Schauder basis of $X$. Using the technique of Fraïssé theory, we construct a rational $K$-based Banach space $\big(\mathbb U_K,(\mathbf e_n)_{n\inω}\big)$ which is $\mathfrak{RI}_K$-universal in the sense that each basis preserving isometry $f:Λ\to\mathbb U_K$ defined on a based subspace $Λ$ of a finite-dimensional rational $K$-based Banach space $A$ extends to a basis preserving isometry $\bar f:A\to\mathbb U_K$ of the based Banach space $A$. We also prove that the $K$-based Banach space $\mathbb U_K$ is almost $\mathfrak{FI}_1$-universal in the sense that any base preserving $\varepsilon$-isometry $f:Λ\to\mathbb U_K$ defined on a based subspace $Λ$ of a finite-dimensional $1$-based Banach space $A$ extends to a base preserving $\varepsilon$-isometry $\bar f:A\to\mathbb U_K$ of the based Banach space $A$. On the other hand, we show that no almost $\mathfrak{FI}_K$-universal based Banach space exists for $K>1$. The Banach space $\mathbb U_K$ is isomorphic to the complementably universal Banach space for the class of Banach spaces with an unconditional Schauder basis, constructed by Pełczyński in 1969.

math.FA↗