General Topology

This article explores the connections between graph-theoretical and topological approaches in the study of the Jordan curve theorem for grids. Building on the foundational work of Rosenfeld, who developed adjacency-based concepts on…

In this paper, ideas of open ball, closed ball, compact set are introduced and some related basic properties are studied. Some topological properties and some other well known results of metric spaces including Cantor intersection theorem…

Let US be the class of all ultrametric spaces generated by labeled star graphs. We prove that compact US-spaces are the completions of totally bounded ultrametric spaces generated by decreasingly labeled rays. We characterize the…

In recent studies, Bibiloni-Femenias, Mi\~{n}ana and Valero characterized the functions that aggregate a family of (quasi-)(pseudo)metric modulars defined over a fixed set $X$ into a single one. In this paper, we adopt a related but…

We consider a new type of mappings in metric spaces so-called mappings contracting total pairwise distance on $n$ points. It is shown that such mappings are continuous. A theorem on the existence of periodic points for such mappings is…

In this paper we investigate Schur ultrafilters on groups. Using the algebraic structure of Stone-\v{C}ech compactifications of discrete groups and Schur ultrafilters, we give a new description of Bohr compactifications of topological…

It is proved that if some boundary $B$ of a convex compact subset $X$ of a locally convex linear space has a countable network, then the convex compact space $X$ is metrizable. If the boundary $B$ is a Lindelof $\Sigma$-space, then the…

We investigate whether the set of subfit elements of a distributive semilattice is an ideal. This question was raised by the second author at the BLAST conference in 2022. We show that in general it has a negative solution, however if the…

We prove that if a separable metrizable $X$ is a union of two disjoint 0-dimensional sets $E$, $F$, $E$ is absolutely $G_{\delta}$ and $F$ is absolutely $F_{\sigma\delta}$ then there is a closed embedding $h$ into the union of countable…

We study topological versions of an independent set in an abelian group and a linearly independent set in a vector space, a {\em topologically independent set} in a topological group and a {\em topologically linearly independent set} in a…

The purpose of this paper is to study more general real-valued functions of two variables than just metrics on a set X. We concentrate mainly on the classes of distances and almost distances. We also introduce the notion of a bridge on the…

The Hurewicz property is a classical generalization of $\sigma$-compactness and Sierpi\'nski sets (whose existence follows from CH) are standard examples of non-$\sigma$-compact Hurewicz spaces. We show, solving a problem stated by Szewczak…

For each countable ordinal $\alpha$, we introduce an ideal $conv_\alpha$ and use it to characterize the class of all compact countable spaces which are homeomorphic to the space $\omega^{\alpha}\cdot n+1$ with the order topology. The…

The notion of hereditarily equivalent continua is classical in continuum theory with only two known nondegenerate examples (arc, and pseudoarc). In this paper we introduce generically hereditarily equivalent continua, i.e. continua which…

We prove that a compact space $K$ embeds into a $\sigma$-product of compact metrizable spaces ($\sigma$-product of intervals) if and only if $K$ is (strongly countable-dimensional) hereditarily metalindel\"of and every subspace of $K$ has a…

We initiate a systematic investigation of group actions on compact medain algebras via the corresponding dynamics on their spaces of measures. We show that a probability measure which is invariant under a natural push forward operation must…

We show that lattice isomorphisms between lattices of slowly oscillating functions on chain-connected proper metric spaces induce coarsely equivalent homeomorphisms. This result leads to a Banach-Stone-like theorem for these lattices.…

Using approximation by continuous functions we prove the following statements to types of tightness in a space $Q_p(X, \mathbb{R})$ of all quasicontinuous real-valued functions with the topology $\tau_p$ of pointwise convergence: the…

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…

Inspired by Zhao and Xu's study on which a dcpo can be determined by its Scott closed subsets lattice, we further investigate whether a poset (or dcpo) $P$ is able to be determined by the family $\mathcal Q(P)$ of its Scott compact…