arXiv subjects

Andrzej Kucharski

Publications and source records attributed to Andrzej Kucharski.

17 recordsLinked to original sources

arXiv2026

Homogeneity of profinite graphs

Profinite graphs are inverse limits of sequences of nonempty finite graphs. There exists a ``generic'' graph $\Pe$ in this category, although its edge relation is rather poor: only isolated vertices and isolated edges. On the other hand, its topological structure (besides being homeomorphic to the Cantor set) is not trivial, as we show in this note. We prove a Knaster--Reichbach type result: Every topological isomorphism $h\colon K\to L$ where $K$ and $L$ are nowhere dense closed subsets of $\Pe$ consisting of isolated vertices can be extended to a topological automorphism of the whole graph $\Pe$.

math.GN↗

arXiv2026

Characterising projective and weakly projective Boolean algebras by coherent Freese--Nation separations

We isolate a coherent finite-separation strengthening of the Freese--Nation property and use it to characterise projective and weakly projective Boolean algebras. A Boolean algebra is projective if and only if it has a meet-closed decomposition base carrying such a coherent finite-separation map; it is weakly projective if and only if it has a meet-closed $π$-base carrying one. The global form of the same property is much stronger: the positive cone of a Boolean algebra carries a coherent finite-separation map if and only if the algebra is countable. Thus the projective and weakly projective characterisations are intrinsically local and cannot be strengthened by requiring the whole algebra to carry the coherent map except in the countable case.

math.LO↗

arXiv2024

On generic topological embeddings

We show that an embedding of a fixed 0-dimensional compact space $K$ into the Čech--Stone remainder $ω^*$ as a nowhere dense P-set is the unique generic limit, a special object in the category consisting of all continuous maps from $K$ to compact metric spaces. Using Fraïssé theory we get a few well know theorems about Čech--Stone remainder. We establish the following: -- an ultrametric space $K$ of weight $κ$ can be uniformly embedded into $κ^κ$ as a uniformly nowhere dense subset, -- every uniform homeomorphism of uniformly nowhere dense sets in $κ^κ$ can be extended to a uniform auto-homeomorphism of $κ^κ$, -- every uniformly nowhere dense set in $κ^κ$ is a uniform retract of $κ^κ$. If we assume that $κ$ is a weakly compact cardinal we get the counterpart of the above result without the uniformity assumption.

math.GN↗

arXiv2022

Scattered $P$-spaces of weight $ω_1$

We examine dimensional types of scattered $P$-spaces of weight $ω_1$. Such spaces can be embedded into $ω_2$. There are established similarities between dimensional types of scattered separable metric spaces and dimensional types of $P$-spaces of weight $ω_1$ with Cantor--Bendixson rank less than $ω_1$.

math.GN↗

arXiv2021

Dimensional types and P-spaces

We investigate the category of discrete topological spaces, with emphasis on inverse systems of height $ω_1$. Their inverse limits belong to the class of $P$-spaces, which allows us to explore dimensional types of these spaces.

math.GN↗

arXiv2019

The Niemytzki plane is kappa-metrizable

We try to explain the differences between the concepts of stratifiable space and $\varkappa$-metrizable space. In particular, we give a characterization of $\varkappa$-metrizable spaces which is modelled on Chigogidze's characterization. Moreover, we present a $\varkappa$-metric for the Niemytzki plane, using the properties of the Euclidean metric.

math.GN↗

arXiv2019

A generalization of $\varkappa$-metrizable spaces

We introduce a new class of $\varkappa$-metrizable spaces, namely countably $\varkappa$-metrizable spaces. We show that the class of all $\varkappa$-metrizable spaces is a proper subclass of counably $\varkappa$-metrizable spaces. On the other hand, for pseudocompact spaces the new class coincides with $\varkappa$-metrizable spaces. We prove a generalization of a Chigogidze result that the Čech-Stone compactification of a pseudocompact countably $\varkappa$-metrizable space is $\varkappa$-metrizable.

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↗

arXiv2016

Universally Kuratowski-Ulam spaces and open-open games

We examine the class of spaces in which the second player has a winning strategy in the open--open game. We show that this spaces are not universally Kuratowski-Ulam. We also show that the games G and G7 introduced by P. Daniels, K. Kunen, H. Zhou [Fund. Math. 145 (1994), no. 3, 205--220] are not equivalent.

math.GN↗

arXiv2015

Game theoretic approach to skeletally Dugundji and Dugundji spaces

Characterizations of skeletally Dugundji spaces and Dugundji spaces are given in terms of club collections, consisting of countable families of co-zero sets. For example, a Tychonoff space $X$ is skeletally Dugundji if and only if there exists an additive $c$-club on $X$. Dugundji spaces are characterized by the existence of additive $d$-clubs.

math.GN↗

arXiv2013

Parovicenko spaces with structures

We study an analogue of the Parovicenko property in categories of compact spaces with additional structures. In particular, we present an internal characterization of this property in the class of compact median spaces.

math.GN↗

arXiv2012

On open-open games of uncountable length

The aim of this note is to investigate the open-open game of uncountable length. We introduce a cardinal number $μ(X)$, which says how long the Player I has to play to ensure a victory. It is proved that $\su(X)\leqμ(X)\leq\su(X)^+$. We also introduce the class $\mathcal C_κ$ of topological spaces that can be represented as the inverse limit of $κ$-complete system $\{X_σ,π^σ_ρ,Σ\}$ with $\w(X_σ)\leqκ$ and skeletal bonding maps. It is shown that product of spaces which belong to $\mathcal C_κ$ also belongs to this class and $μ(X)\leqκ$ whenever $X\in\mathcal C_κ$ .

math.GN↗

arXiv2011

On functors preserving skeletal maps and skeletally generated compacta

A map $f:X\to Y$ between topological spaces is skeletal if the preimage $f^{-1}(A)$ of each nowhere dense subset $A\subset Y$ is nowhere dense in $X$. We prove that a normal functor $F:Comp\to Comp$ is skeletal (which means that $F$ preserves skeletal epimorphisms) if and only if for any open surjective open map $f:X\to Y$ between zero-dimensional compacta with two-element non-degeneracy set $N^f=\{x\in X:|f^{-1}(f(x))|>1\}$ the map $Ff:FX\to FY$ is skeletal. This characterization implies that each open normal functor is skeletal. The converse is not true even for normal functors of finite degree. The other main result of the paper says that each normal functor $F: Comp\to Comp$ preserves the class of skeletally generated compacta. This contrasts with the known Shchepin's result saying that a normal functor is open if and only if it preserves openly generated compacta.

math.GN↗

arXiv2011

A spectral characterization of skeletal maps

We prove that a map between two realcompact spaces is skeletal if and only if it is homeomorphic to the limit map of a skeletal morphism between w-spectra with surjective limit projections.

math.GN↗

arXiv2008

Inverse Systems and I-Favorable Spaces

A compact space X is I-favorable if, and only if X can be representing as a limit of $σ$-complete inverse system of compact metrizable spaces with skeletal bonding maps.

math.GN↗

arXiv2006

Cardinal invariants for C-cross topologies

C-cross topologies are introduced. Modifcations of the Kuratowski-Ulam Theorem are considered. Cardinal invariants add, cof, cov and non with respect to meager or nowhere dense subsets are compared. Remarks on invariants cof(nwdY) are mentioned for dense subspaces Y of X.

math.GN↗