General Topology

For Erd\H{o}s space, $\mathfrak{E}$, let us define a topology, $\tau_{clopen}$, which is generated by clopen subsets of $\mathfrak{E}$. A. V. Arhangel'skii and J. Van Mill asked whether the topology $\tau_{clopen}$ is compatible with the…

For every finite closure space $X$ one can define a finite topological space $\operatorname{Top} X$ together with a natural projection $\operatorname{Top} X\longrightarrow X$. This could allow to apply the techniques of topological…

It is established that any homeomorphism between two closed negligible subset of $D^\tau$ can be extended to an autohomeomorphism of $D^\tau$.

We study the problem of topologically order-embedding a given topological poset X in the space of all closed subsets of X which is topologized by the Fell topology and ordered by set inclusion. We show that this can be achieved whenever X…

For a Hausdorff zero-dimensional topological space $X$ and a totally ordered field $F$ with interval topology, let $C_c(X,F)$ be the ring of all $F-$valued continuous functions on $X$ with countable range. It is proved that if $F$ is either…

For a topological space $X$, let $CL(X)$ be the set of all non-empty closed subset of $X$, and denote the set $CL(X)$ with the Vietoris topology by $(CL(X), \mathbb{V})$. In this paper, we mainly discuss the hyperspace $(CL(X), \mathbb{V})$…

Here we have introduced and studied the idea of $ Ig^*$-closed set with respect to an ideal and investigated some of its properties in Alexandroff spaces. We have also introduced $ Ig^*$-$T_0 $ axiom, $ Ig^*$-$T_1$ axiom, $ Ig^*$-$T_\omega…

A space $X$ is $D$ if for every assignment, $U$, of an open neighborhood to each point $x$ in $X$ there is a closed discrete $D$ such that $\bigcup \{U(x) : x \in D\}=X$. The box product, $\square X^\omega$, is $X^\omega$ with topology…

In this paper, we first define the concept of convexity in G-metric spaces. We then use Mann iterative process in this newly defined convex G-metric space to prove some convergence results for some classes of mappings. In this way, we can…

In this paper, we deal with uniform spaces whose diagonal uniformity admits a basis consisting of equivalence relations. Such non-Archimedean uniform spaces are particularly interesting for applications in commutative ring theory, because…

In this paper, it is proved that every topological gyrogroup $G$ is topologically groupoid isomorphic to a closed subgyrogroup of a connected, locally connected topological gyrogroup $G^{\bullet}$.

This thesis belongs to the area of General Topology and, in particular, to the field of study of uniform spaces. It is divided in three parts where the related topics Bourbaki-completeness and Samuel realcompactification are studied. In the…

In this paper we collect some open set-theoretic problems that appear in the large-scale topology (called also Asymptology). In particular we ask problems about critical cardinalities of some special (large, indiscrete, inseparated) coarse…

In this paper, we prove that in forcing extensions by a poset with finally property K over a model of GCH+$\square$, every compact sequentially compact space is weakly pseudoradial. We also prove the following assuming $\mathfrak{s}\leq…

Let $X$ be separable metrizable, and let $f\subseteq X^2$ be a non-trivial relation on $X$. For a given partial order $(P,\leq)$, the Mahavier product $M(X,f,P)\subseteq X^P$ (also known as a generalized inverse limit) collects functions…

A bornology $\mathcal{B}$ on a set $X$ is called minmax if the smallest and the largest coarse structures on $X$ compatible with $\mathcal{B}$ coincide. We prove that $\mathcal{B}$ is minmax if and only if the family $\mathcal…

Let $\kappa$ be an infinite cardinal. A topological space $X$ is $\kappa$-bounded if the closure of any subset of cardinality $\le\kappa$ in $X$ is compact. We discuss the problem of embeddability of topological spaces into Hausdorff…

For a Tychonoff space $X$ and a subspace $Y\subset\mathbb R$, we study Baire category properties of the space $C_{\downarrow F}(X,Y)$ of continuous functions from $X$ to $Y$, endowed with the Fell hypograph topology. We characterize pairs…

We study the interplay between three weak topologies on a topological semilattice $X$: the weak$^\circ$ topology $\mathcal W^\circ_X$ (generated by the base consiting of open subsemilattices of $X$), the weak$^\bullet$ topology $\mathcal…

In the paper we present various characterizations of chain-compact and chain-finite topological semilattices. A topological semilattice $X$ is called chain-compact (resp. chain-finite) if each closed chain in $X$ is compact (finite). In…