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W. Kubiś

Publications and source records attributed to W. Kubiś.

9 recordsLinked to original sources
arXiv2023

Homogeneous ultrametric structures

We develop the theory of homogeneous Polish ultrametric structures. Our starting point is a Fraisse class of finite structures and the crucial tool is the universal homogeneous epimorphism. The new Fraisse limit is an inverse limit, nevertheless its universality is with respect to embeddings and, contrary to the Polish metric Fraisse theory of Ben Yaacov, homogeneity is strict. Our development can be viewed as the third step of building a Borel-like hierarchy of Fraisse limits, where the first step was the classical setting of Fraisse and the second step is the more recent theory, due to Irwin and Solecki, of pro-finite Fraisse limits.

math.LO↗
arXiv2023

On the connectivity of graph Lipscomb's space

A central role in topological dimension theory is played by Lipscomb's space $J_{A}$ since it is a universal space for metric spaces of weight $|A|\geq \aleph _{0}$. On the one hand, Lipscomb's space is the attractor of a possibly infinite iterated function system, i.e. it is a generalized Hutchinson-Barnsley fractal. As, on the other hand, some classical fractal sets are universal spaces, one can conclude that there exists a strong connection between topological dimension theory and fractal set theory. A generalization of Lipscomb's space, using graphs, has been recently introduced (see R. Miculescu, A. Mihail, Graph Lipscomb's space is a generalized Hutchinson-Barnsley fractal, Aequat. Math., \textbf{96} (2022), 1141-1157). It is denoted by $J_{A}^{\G}$ and it is called graph Lipscomb's space associated with the graph $\G$ on the set $A$. It turns out that it is a topological copy of a generalized Hutchinson-Barnsley fractal. This paper provides a characterization of those graphs $\G$ for which $J_{A}^{\G}$ is connected. In the particular case when $A$ is finite, some supplementary characterizations are presented.

math.GN↗
arXiv2021

Vietoris hyperspaces of scattered Priestley spaces

We study Vietoris hyperspaces of closed and closed final sets of Priestley spaces. We are particularly interested in Skula topologies. A topological space is \emph{Skula} if its topology is generated by differences of open sets of another topology. A compact Skula space is scattered and moreover has a natural well-founded ordering compatible with the topology, namely, it is a Priestley space. One of our main objectives is investigating Vietoris hyperspaces of general Priestley spaces, addressing the question when their topologies are Skula and computing the associated ordinal ranks. We apply our results to scattered compact spaces based on certain almost disjoint families, in particular, Lusin families and ladder systems.

math.GN↗
arXiv2019

Homogeneous probability measures on the Cantor set

We show that every homeomorphism between closed measure zero subsets extends to a measure preserving auto-homeomorphism, whenever the Cantor set is endowed with a suitable probability measure. This is valid both for the standard product measure, as well as for the universal homogeneous rational measure.

math.PR↗
arXiv2017

Game-theoretic characterization of the Gurarii space

We present a simple and natural infinite game building an increasing chain of finite-dimensional Banach spaces. We show that one of the players has a strategy with the property that, no matter how the other player plays, the completion of the union of the chain is linearly isometric to the Gurarii space.

math.FA↗
arXiv2016

A separable Fréchet space of almost universal disposition

The Gurari\uı space is the unique separable Banach space $\mathbb{G}$ which is of almost universal disposition for finite-dimensional Banach spaces, which means that for every $\varepsilon>0$, for all finite-dimensional normed spaces $E \subseteq F$, for every isometric embedding ${e}\colon{E}\to{\mathbb{G}}$ there exists an $\varepsilon$-isometric embedding ${f}\colon{F}\to{\mathbb{G}}$ such that $f \restriction E = e$. We show that $\mathbb{G}^{\mathbb{N}}$ with a special sequence of semi-norms is of almost universal disposition for finite-dimensional graded Fréchet spaces. The construction relies heavily on the universal operator on the Gurari\uı space, recently constructed by Garbulińska-Wegrzyn and the third author. This yields in particular that $\mathbb{G}^{\mathbb{N}}$ is universal in the class of all separable Fréchet spaces.

math.FA↗
arXiv2014

Networks for the weak topology of Banach and Fréchet spaces

We start the systematic study of Fréchet spaces which are $\aleph$-spaces in the weak topology. A topological space $X$ is an $\aleph_0$-space or an $\aleph$-space if $X$ has a countable $k$-network or a $σ$-locally finite $k$-network, respectively. We are motivated by the following result of Corson (1966): If the space $C_{c}(X)$ of continuous real-valued functions on a Tychonoff space $X$ endowed with the compact-open topology is a Banach space, then $C_{c}(X)$ endowed with the weak topology is an $\aleph_0$-space if and only if $X$ is countable. We extend Corson's result as follows: If the space $E:=C_{c}(X)$ is a Fréchet lcs, then $E$ endowed with its weak topology $σ(E,E')$ is an $\aleph$-space if and only if $(E,σ(E,E'))$ is an $\aleph_0$-space if and only if $X$ is countable. We obtain a necessary and some sufficient conditions on a Fréchet lcs to be an $\aleph$-space in the weak topology. We prove that a reflexive Fréchet lcs $E$ in the weak topology $σ(E,E')$ is an $\aleph$-space if and only if $(E,σ(E,E'))$ is an $\aleph_0$-space if and only if $E$ is separable. We show however that the nonseparable Banach space $\ell_{1}(\mathbb{R})$ with the weak topology is an $\aleph$-space.

math.FA↗
arXiv2004

Semi-Eberlein spaces

We investigate the class of compact spaces which are embeddable into a power of the real line $R^κ$ in such a way that c_0(κ) is dense in the image. We show that this is a proper subclass of the class of Valdivia, even when restricted to Corson compacta. We prove a preservation result concerning inverse sequences with semi-open retractions. As a corollary we obtain that retracts of Cantor or Tikhonov cubes belong to the above class.

math.GN↗