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Yaryna Stelmakh

Publications and source records attributed to Yaryna Stelmakh.

6 recordsLinked to original sources
arXiv2023

Examples of strongly rigid countable (semi)Hausdorff spaces

A topological space $X$ is $strongly$ $rigid$ if each non-constant continuous map $f:X\to X$ is the identity map of $X$. A Hausdorff topological space $X$ is called $Brown$ if for any nonempty open sets $U,V\subseteq X$ the intersection $\bar U\cap\bar V$ is infinite. We prove that every second-countable Brown Hausdorff space $X$ admits a stronger topology $τ'$ such that $X'=(X,τ')$ is a strongly rigid anticompact Brown space.This construction yields an example of a countable anticompact Hausdorff space $X$ which is strongly rigid, which answers two problems posed at MathOverflow. By the same method we construct a strongly rigid $k_2$-metrizable semi-Hausdorff space containing a non-closed compact subset, which answers two other problem posed at MathOverflow.

math.GN↗
arXiv2020

Homeomorphisms of the space of non-zero integers with the Kirch topology

The $Golomb$ (resp. $Kirch$) topology on the set $\mathbb Z^\bullet$ of nonzero integers is generated by the base consisting of arithmetic progressions $a+b\mathbb Z=\{a+bn:n\in\mathbb Z\}$ where $a\in\mathbb Z^\bullet$ and $b$ is a (square-free) number, coprime with $a$. In 2019 Dario Spirito proved that the space of nonzero integers endowed with the Golomb topology admits only two self-homeomorphisms. In this paper we prove an analogous fact for the space of nonzero integers endowed with the Kirch topology: it also admits exactly two self-homeomorphisms.

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↗
arXiv2020

The cometrizability of generalized metric spaces

A topological space $X$ is cometrizable if it admits a weaker metrizable topology such that each point $x\in X$ has a (not necessarily open) neighborhood base consisting of metrically closed sets. We study the relation of cometrizable spaces to other generalized metric spaces and prove that all $\mathsf{as}$-cosmic spaces are cometrizable. Also, we present an example of a regular countable space of weight $ω_1$, which is not cometrizable. Under $ω_1=\mathfrak c$ this space contains no infinite compact subsets and hence is $\mathsf{cs}$-cosmic. Under $ω_1<\mathfrak p$ this countable space is Fréchet-Urysohn and is not $\mathsf{cs}$-cosmic.

math.GN↗
arXiv2020

A universal coregular countable second-countable space

A Hausdorff topological space $X$ is called $\textit{superconnected}$ (resp. $\textit{coregular}$) if for any nonempty open sets $U_1,\dots U_n\subseteq X$, the intersection of their closures $\bar U_1\cap\dots\cap\bar U_n$ is not empty (resp. the complement $X\setminus (\bar U_1\cap\dots\cap\bar U_n)$ is a regular topological space). A canonical example of a coregular superconnected space is the projective space $\mathbb Q\mathsf P^\infty$ of the topological vector space $\mathbb Q^{<ω}=\{(x_n)_{n\inω}\in \mathbb Q^ω:|\{n\inω:x_n\ne 0\}|<ω\}$ over the field of rationals $\mathbb Q$. The space $\mathbb Q\mathsf P^\infty$ is the quotient space of $\mathbb Q^{<ω}\setminus\{0\}^ω$ by the equivalence relation $x\sim y$ iff $\mathbb Q{\cdot}x=\mathbb Q{\cdot}y$. We prove that every countable second-countable coregular space is homeomorphic to a subspace of $\mathbb Q\mathsf P^\infty$, and a topological space $X$ is homeomorphic to $\mathbb Q\mathsf P^\infty$ if and only if $X$ is countable, second-countable, and admits a decreasing sequence of closed sets $(X_n)_{n\inω}$ such that (i) $X_0=X$, $\bigcap_{n\inω}X_n=\emptyset$, (ii) for every $n\inω$ and a nonempty open set $U\subseteq X_n$ the closure $\bar U$ contains some set $X_m$, and (iii) for every $n\inω$ the complement $X\setminus X_n$ is a regular topological space. Using this topological characterization of $\mathbb Q\mathsf P^\infty$ we find topological copies of the space $\mathbb Q\mathsf P^\infty$ among quotient spaces, orbit spaces of group actions, and projective spaces of topological vector spaces over countable topological fields.

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