arXiv subjects

Sławomir Turek

Publications and source records attributed to Sławomir Turek.

9 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↗

arXiv2020

The Kirch space is topologically rigid

The $Golomb$ $space$ (resp. the $Kirch$ $space$) is the set $\mathbb N$ of positive integers endowed with the topology generated by the base consisting of arithmetic progressions $a+b\mathbb N_0=\{a+bn:n\ge 0\}$ where $a\in\mathbb N$ and $b$ is a (square-free) number, coprime with $a$. It is known that the Golomb space (resp. the Kirch space) is connected (and locally connected). By a recent result of Banakh, Spirito and Turek, the Golomb space has trivial homeomorphism group and hence is topologically rigid. In this paper we prove the topological rigidity of the Kirch space.

math.GN↗

arXiv2019

The Golomb space is topologically rigid

The $Golomb$ $space$ $\mathbb N_τ$ is the set $\mathbb N$ of positive integers endowed with the topology $τ$ generated by the base consisting of arithmetic progressions $\{a+bn:n\ge 0\}$ with coprime $a,b$. We prove that the Golomb space $\mathbb N_τ$ is topologically rigid in the sense that its homeomorphism group is trivial. This resolves a problem posed by the first author at Mathoverflow in 2017.

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

Supercompact minus compact is super

According to a folklore characterization of supercompact spaces, a compact Hausdorff space is supercompact if and only if it has a binary closed $k$-network. This characterization suggests to call a topological space $super$ if it has a binary closed $k$-network $\mathcal N$. The binarity of $\mathcal N$ means that every linked subfamily of $\mathcal N$ is centered. Therefore, a Hausdorff space is supercompact if and only if it is super and compact. In this paper we prove that the class of super spaces contains all GO-spaces, all supercompact spaces, all metrizable spaces, and all collectionwise normal $\aleph$-spaces. Moreover, the class of super spaces is closed under taking Tychonoff products and discretely dense sets in Tychonoff products. The superness of metrizable spaces implies that each compact metrizable space is supercompact, which was first proved by Strok and Szymański (1975) and then reproved by Mills (1979), van Douwen (1981), and D{\c e}bski (1984).

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↗

arXiv2010

A decomposition theorem for compact groups with application to supercompactness

We show that every compact connected group is the limit of a continuous inverse sequence, in the category of compact groups, where each successor bonding map is either an epimorphism with finite kernel or the projection from a product by a simple compact Lie group. As an application, we present a proof of an unpublished result of Charles Mills from 1978: every compact group is supercompact.

math.GN↗