General Topology

This survey presents some historical background and recent developments in the area of selections for set-valued mappings along with several open questions. It was written with the hope that the presented material may pique an interest in…

Here we have studied on the ideas of $g_{\mu_i}$ and $\lambda_{\mu_i}$-closed sets with respect to ${\mu_j}(i,j=1,2,i\not=j)$ and pairwise $ \lambda $-closed sets in a generalized bitopological space $ (X,\mu_1, \mu_2) $. We have also…

Under $\mathfrak{p} = \mathfrak{c}$, we answer Question 24 of \cite{dikranjan&shakhmatov3} for cardinality ${\mathfrak c}$ , by showing that if a non-torsion Abelian group of size continuum admits a countably compact Hausdorff group…

A space has $\sigma$-compact tightness if the closures of $\sigma$-compact subsets determines the topology. We consider a dense set variant that we call densely k-separable. We consider the question of whether every densely k-separable…

For a Tychonoff space $X$ and a family $\lambda$ of subsets of $X$, we denote by $C_{\lambda}(X)$ the $T_1$-space of all real-valued continuous functions on $X$ with the $\lambda$ -open topology. A topological space is productively…

In this paper we show that polycyclic monoids are universal objects in the class of graph inverse semigroups. In particular, we prove that a graph inverse semigroup $G(E)$ over a directed graph $E$ embeds into the polycyclic monoid…

Given a family of continuous real functions $\mathcal{G}$, let $R_\mathcal{G}$ be a binary relation defined as follows: a continuous function $f\colon\mathbb{R}\to\mathbb{R}$ is in the relation with a closed set $E\subseteq\mathbb{R}$ if…

This document presents the proof that the epimorphisms of the category of Hausdorff spaces are exactly the image dense morphisms. While it is a classical result; its proof is difficult to find in internet. Consequently, I decided to write…

Often, a given selection game studied in the literature has a known dual game. In dual games, a winning strategy for a player in either game may be used to create a winning strategy for the opponent in the dual. For example, the Rothberger…

For the algebraic convergence $\lambda_{\mathrm{s}}$, which generates the well known sequential topology $\tau_s$ on a complete Boolean algebra ${\mathbb B}$, we have $\lambda_{\mathrm{s}}=\lambda_{\mathrm{ls}}\cap \lambda_{\mathrm{li}}$,…

For a Tychonoff space $X$, we will denote by $USC_{p}(X)$ ($B_1(X)$) a set of all real-valued upper semicontinuous functions (a set of all Baire functions of class 1) defined on $X$ endowed with the pointwise convergence topology. In this…

We compare the forcing related properties of a complete Boolean algebra B with the properties of the convergences $\lambda_s$ (the algebraic convergence) and $\lambda_{ls}$ on B generalizing the convergence on the Cantor and Aleksandrov…

We investigate the possibility of extension of fragmented functions from Lindel\"{o}f subspaces of completely regular spaces and find necessary and sufficient conditions on a fragmented Baire-one function to be extendable on any completely…

In accordance with $M_3$-structures in paper [4], we construct a stratifiable space which is not $M_1$-spaces.

We prove that in the Miller model the Menger property is preserved by finite products of metrizable spaces. This answers several open questions and gives another instance of the interplay between classical forcing posets with fusion and…

Let $X$ be a partially ordered set with the property that each family of order intervals of the form $[a,b],[a,\rightarrow )$ with the finite intersection property has a nonempty intersection. We show that every directed subset of $X$ has a…

Non-Newtonian calculus that starts with elementary non-Diophantine arithmetic operations of a Burgin type is applicable to all fractals whose cardinality is continuum. The resulting definitions of derivatives and integrals are simpler from…

In this paper we will prove that all the elements in the smallest ideal K($\beta$N) in the semigroup of the Stone Cech compactification ($\beta$N,.) of the discrete semigroup of natural numbers N under multiplication constitute a single…

We look at the Bohr topology of maximally almost periodic groups (MAP, for short). Among other results, we investigate when a totally bounded abelian group $(G,w)$ is the Bohr reflection of a locally compact abelian group. Necessary and…

The countable uniform power (or uniform box product) of a uniform space $X$ is a special topology on ${}^{\omega}X$ that lies between the Tychonoff topology and the box topology. We solve an open problem posed by P. Nyikos showing that if…