Related papers: On topological spaces and topological groups with …

we prove that if $X$ is a locally compact $\sigma$-compact space then on its quotient, $\gamma(X)$ say, determined by the algebra of all real valued bounded continuous functions on $X$, the quotient topology and the completely regular…

For a Tychonoff space $X$, we denote by $(C(X), \tau_k, \tau_p)$ the bitopological space of all real-valued continuous functions on $X$ where $\tau_k$ is the compact-open topology and $\tau_p$ is the topology of pointwise convergence. In…

Let $\FP(X)$ be the free paratopological group on a topological space $X$ in the sense of Markov. In this paper, we study the group $\FP(X)$ on a $P_\alpha$-space $X$ where $\alpha$ is an infinite cardinal and then we prove that the group…

We investigate the local topological structure of non-metrizable topological groups through the lens of Tukey order and cofinal types. Motivated by recent advances in topological groups admitting an $\omega^\omega$-base, we introduce the…

We prove a pointwise ergodic theorem for quasi-probability-measure-preserving (quasi-pmp) locally countable measurable graphs, equivalently, Schreier graphs of quasi-pmp actions of countable groups. For ergodic graphs, the theorem gives an…

We show that if $Y$ is a dense subspace of a Tychonoff space $X$, then $w(X)\leq nw(Y)^{Nag(Y)}$, where $Nag(Y)$ is the Nagami number of $Y$. In particular, if $Y$ is a Lindel\"of $\Sigma$-space, then $w(X)\leq nw(Y)^\omega\leq…

In this paper, we mainly prove that if $H$ is a closed strong subgyrogroup of a strongly topological gyrogroup $G$ and $H$ is neutral, then (1) $G/H$ is biradial if and only if $G/H$ is nested; (2) $G/H$ is metrizable if and only if $G/H$…

In this paper we introduce and study three new cardinal topological invariants called the cs*, cs-, and sb-characters. The class of topological spaces with countable cs*-character is closed under many topological operations and contains all…

We construct some new cohomology theories for topological groups and Lie groups and study some of its basic properties. For example, we introduce a cohomology theory based on measurable cochains which are continuous in a neighbourhood of…

We call a space $X$ {\it weakly linearly Lindel\"of} if for any family $\mathcal{U}$ of non-empty open subsets of $X$ of regular uncountable cardinality $\kappa$, there exists a point $x\in X$ such that every neighborhood of $x$ meets…

By analogy with the Cayley graph of a group with respect to a finite generating set or the Cayley--Abels graph of a totally disconnected, locally compact group, we detail countable connected graphs associated to Polish groups that we term…

The deck of a topological space $X$ is the set $\mathcal{D}(X)=\{[X \setminus \{x\}] \colon x \in X\}$, where $[Z]$ denotes the homeomorphism class of $Z$. A space $X$ is topologically reconstructible if whenever…

A base $\mathcal{B}$ for a space $X$ is said to be sharp if, whenever $x\in X$ and $(B_n)_{n\in\omega}$ is a sequence of pairwise distinct elements of $\mathcal{B}$ each containing $x$, the collection $\{\bigcap_{j\le n}B_j:n\in\omega\}$ is…

We analyze the effective content of countable, second countable topological spaces by directly calculating the complexity of several topologically defined index sets. We focus on the separation principles, calibrating an arithmetic…

We study free topological groups defined over uniform spaces in some subclasses of the class NA of non-archimedean groups. Our descriptions of the corresponding topologies show that for metrizable uniformities the corresponding free…

Some new classes of compacta $K$ are considered for which $C(K)$ endowed with the pointwise topology has a countable cover by sets of small local norm--diameter.

We study when a piecewise full group (a.k.a. topological full group) of homeomorphisms of the Cantor space $X$ can be given a non-discrete totally disconnected locally compact (t.d.l.c.) topology and give a criterion for the alternating…

All spaces are assumed to be Tychonoff. Given a realcompact space $X$, we denote by $\mathsf{Exp}(X)$ the smallest infinite cardinal $\kappa$ such that $X$ is homeomorphic to a closed subspace of $\mathbb{R}^\kappa$. Our main result shows…

A topological space $X$ is Baire if the intersection of any sequence of open dense subsets of $X$ is dense in $X$. One of the interesting problems for the space of Baire functions is the Banakh-Gabriyelyan problem: Let $\alpha$ be a…

This article fits in the context of the approach to topological problems in terms of the underlying convergence space structures, and serves as yet another illustration of the power of the method. More specifically, we spell out…