General Topology
We prove that every free group G with infinitely many generators admits a Hausdorff group topology T with the following property: for every T-open neighbourhood U of the identity of G, each element g in G can be represented as a product…
It is given a simplified and self-contained proof of the classical Michael's finite-dimensional selection theorem. The proof is based on approximate selections constructed stepwise over skeletons of nerves of covers. The method is also…
It is introduced the concept of a quasi-king space, which is a natural generalisation of a king space. In the realm of suborderable spaces, king spaces are precisely the compact spaces, so are the quasi-king spaces. In contrast, quasi-king…
In this paper, we examine some properties of the fixed point set of a digitally continuous function. The digital setting requires new methods that are not analogous to those of classical topological fixed point theory, and we obtain results…
A topological space is defined to be banalytic (resp. analytic) if it is the image of a Polish space under a Borel (resp. continuous) map. A regular topological space is analytic if and only if it is banalytic and cosmic. Each (regular)…
The contour of a family of filters along a filter is a set-theoretic lower limit. Topologicity and regularity of convergences can be characterized with the aid of the contour operation. Contour inversion is studied, in particular, for…
We prove that the cardinality of transitive quasi-uniformities in a quasi-proximity class is at least $2^{2^{\aleph_0}}$ if there exist at least two transitive quasi-uniformities in the class. The transitive elements of $\pi(\delta)$ are…
The aim of this paper is to introduce and investigate a new class of functions called weakly almost contra-$T^*$-continuity which is defined as a function from an operator topological space $(X, \tau, T)$ into an arbitrary topological space…
A metric space $\mathbf{X}$ is called densely complete if there exists a dense set $D$ in $\mathbf{X}$ such that every Cauchy sequence of points of $D $ converges in $\mathbf{X}$. One of the main aims of this work is to prove that the…
The goal of this report is to investigate the variety of Hausdorff compactifications of $\mathbb{R}$. The Alexandroff one-point compactification, the two-point compactification, and the Stone-Cech compactification are all clearly different.…
We prove that if $X$ is a paracompact space, $Y$ is a metric space and $f:X\to Y$ is a functionally fragmented map, then (i) $f$ is $\sigma$-discrete and functionally $F_\sigma$-measurable; (ii) $f$ is a Baire-one function, if $Y$ is weak…
A ballean (or coarse space) is a set endowed with a coarse structure. A ballean $X$ is called normal if any two asymptotically disjoint subsets of $X$ are asymptotically separated. We say that a ballean $X$ is ultranormal (extremely normal)…
Boolean spaces with internal semigroups generalize profinite semigroups and are pertinent for the recognition of not-necessarily regular languages. Via recognition, the study of existential quantification in logic on words amounts to the…
Let $X$ be a uniform space and $U(X)$ the linear space of real-valued uniformly continuous functions on $X$. Our main objective is to give a number of properties characterizing the fact that $U(X)$ is stable under pointwise product in case…
The weak tightness $wt(X)$ of a space $X$ was introduced in [11] with the property $wt(X)\leq t(X)$. We investigate several well-known results concerning $t(X)$ and consider whether they extend to the weak tightness setting. First we give…
A ballean is a set endowed with a coarse structure. We introduce and explore three constructions of balleans from a pregiven family of balleans: bornological products, bouquets and combs. We analyze the smallest and the largest coarse…
A topological space $X$ is $\kappa$-Fr\'{e}chet--Urysohn if for every open subset $U$ of $X$ and every $x\in \overline{U}$ there exists a sequence in $ U$ converging to $x$. We prove that every $\kappa$-Fr\'{e}chet--Urysohn Tychonoff space…
We established order-preserving versions of the basic principles of functional analysis such as Hahn-Banach, Banach-Steinhaus, open mapping and Banach-Alaoglu theorems.
We give a modern exposition and an elementary proof of the topological equivalence between periodic homeomorphisms of the disc and the sphere and euclidean isometries.
A vector balleans is a vector space over $\mathbb{R}$ endowed with a coarse structure in such a way that the vector operations are coarse mappings. We prove that, for every ballean $(X, \mathcal{E})$, there exists the unique free vector…