General Topology

Let $G=(V,E)$ be a finite connected graph with vertex set $V$ and edge set $E$, and let $U(G)$ be the set of all ultrametric spaces $(V,d_l)$ generated by vertex labelings $l\colon V \to \mathbb R^+$. We prove that the inequality $$ |D(V)|…

In 1992, Diestel asked which topological spaces could be represented as the end space of some graph. In 2023, Pitz provided a solution to this question by giving a topological characterization of end spaces using a hereditarily complete…

In this paper, we study homotopical analogues of the Cantor-Bendixson derivative. For each $n\geq 0$, the "$\pi_n$-wild set" $\mathbf{w}_n(X)$ of a topological space $X$ is the subspace of $X$ consisting of the points at which there exists…

In this note a criterion for Cauchy sequences is proved which refines the one presented in `Cauchy sequences in b-metric spaces', Topology Appl. 373 (2025) 109477.

This paper deals with the extension of partial actions of topological groups on topological spaces. Within this framework, we introduce a class of topological embeddings defined via the inverse semigroup of homeomorphisms between open…

In this paper, we introduce the notion of a $\gamma$-density point for Lebesgue-measurable subsets of $\mathbb{R}$, where $\gamma$ is a modulus function, and study its basic measure-theoretic properties. We show that every $\gamma$-density…

Traditional risk measures in finance, predominantly based on the second moment of return distributions or tail risk heuristics (VaR/CVaR), fail to account for the intrinsic geometric structure of market dynamics. This paper introduces a…

A fan is an arc-wise connected hereditarily unicoherent continuum with exactly one branching point. By a result of Borsuk, every fan is a 1-dimensional continuum that can be expressed as the union of a family of arcs, each pair of which…

Topology may be interpreted as the study of verifiability, where opens correspond to semi-decidable properties. In this paper we make a distinction between verifiable properties themselves and processes which carry out the verification…

For any compact set $K$ lying on a closed surface $\mathcal{S}$ we introduce a closed equivalence relation $\sim$, called the {\em Sch\"onflies equivalence} on $K$. We show that every class $[x]_\sim$ of $\sim$ is a continuum and that the…

Let c denote the cardinality of the continuum. Let L denote the family of all Hausdorff topologies on the real line coarser than the natural topology. We construct 2^c pairwise non-homeomorphic completely normal topologies in L among which…

In this paper, we investigate the set $\mathcal{U}(\mathbb{U})$ of universal and ultrahomogeneous $1$-Lipschitz retractions acting on the Urysohn space as the subspace of the space $\mathcal{R}(\mathbb{U})$ of all $1-$Lipschitz retractions…

We introduce a family of forcing notions that are helpful in showing that certain graphs do not have countable colourings of (additive) Borel class alpha. We construct graphs that are ''weakly minimal'' for such colourings.

Following Will Brian, we define a metric space $X$ to be $Banakh$ if all nonempty spheres of positive radius $r$ in $X$ have cardinality $2$ and diameter $2r$. Standard examples of Banakh spaces are subgroups of the real line. In this paper…

We prove that locally countably-compact Hausdorff topological groups $\mathbb{G}$ act continuously on their iterated joins $E_n\mathbb{G}:=\mathbb{G}^{*(n+1)}$ (the total spaces of the Milnor-model $n$-universal $\mathbb{G}$-bundles) as…

We prove that: I. The product of any two regular isodyne spaces of cardinality $\omega_1$ is $\omega$-resolvable; II. The product of any $n + 2$ Hausdorff isodyne spaces of cardinality $\omega_n$ is $\omega$-resolvable.

In answer to a question on Mathoverflow we show that the Boolean algebra $\mathcal{P}(\omega)/\mathit{fin}$ contains a family $\{\mathcal{B}_X:X\subseteq\mathfrak{c}\}$ of subalgebras with the property that $X\subseteq Y$ implies…

The notion of $\ast$-measure on a compact Hausdorff space can be defined for arbitrary continuous triangular norm $\ast$. The well-known Hutchinson-Barnsley theory deals with the iterated function systems (IFSs) of probability measures and…

The paper is devoted to a categorical study of the category of probabilistic metric spaces. The study is based on an isomorphic description of the category of probabilistic metric spaces. The isomorphic description was obtained in [3] and…

We study rings of real-valued continuous functions in terms of pseudocomplementation conditions on various lattices attached to their prime spectrum. We fully characterize pseudocomplementation in all cases and have an almost complete…