General Topology
The paper is devoted to the fixed point theory in four aspects: of contractions, nonexpansive mappings, generalized inward mappings, and of the tool theorems. The manuscript was written about ten years ago. At first Nadler's concept of…
A group topology is said to be linear if open subgroups form a base of neighborhoods of the identity element. It is proved that the existence of a nondiscrete extremally disconnected group of Ulam nonmeasurable cardinality with linear…
Quotient space is a class of the most important topological spaces in the research of topology. In this paper, we show that if G is a strongly topological gyrogroup with a symmetric neighborhood base U at 0 and H is an admissible…
The concept of topological gyrogroups is a generalization of a topological group. In this work, ones prove that a topological gyrogroup G is metrizable iff G has an {\omega}{\omega}-base and G is Frechet-Urysohn. Moreover, in topological…
Given an arbitrary measurable cardinal $\kappa$, a nondiscrete Hausdorff extremally disconnected topological group of cardinality $\kappa$ is constructed.
Assuming the existence of $\mathfrak c$ incomparable selective ultrafilters, we classify the non-torsion Abelian groups of cardinality $\mathfrak c$ that admit a countably compact group topology. We show that for each $\kappa \in [\mathfrak…
The concept of gyrogroups is a generalization of groups which do not explicitly have associativity. Recently, Wattanapan et al consider the construction of Hartman-Mycielski in strongly topological gyrogroups. In this paper, we extend their…
In our paper [18] we showed that a Tychonoff space $X$ is a $\Delta$-space (in the sense of [20], [30]) if and only if the locally convex space $C_{p}(X)$ is distinguished. Continuing this research, we investigate whether the class $\Delta$…
The thesis presents the subject of synthetic topology, especially with relation to metric spaces. A model of synthetic topology is a categorical model in which objects possess an intrinsic topology in a suitable sense, and all morphisms are…
To any compact $K\subset\hat{\mathbb{C}}$ we associate a map $\lambda_K: \hat{\mathbb{C}}\rightarrow\mathbb{N}\cup\{\infty\}$ -- the lambda function of $K$ -- such that a planar continuum $K$ is locally connected if and only if…
If $X$ is a finite tree and $f \colon X \longrightarrow X$ is a map, as the Main Theorem of this paper we find eight conditions, each of which is equivalent to the fact that $f$ is equicontinuous. To name just a few of the results obtained:…
We are concerned with the question of when a homeomorphism between $C_p(X\times \tau)$ and $C_p(Y\times \tau)$ implies the existence of a homeomorphism between $C_p(X\times \tau')$ and $C_p(Y\times \tau')$, where $\tau$ and $\tau'$ are…
Some known fixed point theorems for nonexpansive mappings in metric spaces are extended here to the case of primitive uniform spaces. The reasoning presented in the proofs seems to be a natural way to obtain other general results.
The classical Banach-Mazur game is directly related to the Baire property and the property of being a productively Baire space. In this paper, we discuss two variations of this classic game that are even more related to these properties.
We define the smooth Lipschitz topology on the moduli space and show that each conformal class is dense in the moduli space endowed with Gromov-Hausdorff topology, which offers an answer to the Tuschmann's question.
In this article, we consider $\mathcal{I}^\mathcal{K}$-convergence to define a new concept of convergence namely, $\mathcal{S}$-$\mathcal{I}^\mathcal{K}$-convergence which generalizes the notion of $\mathcal{S}$-$\mathcal{I}$-convergence…
The weak tightness $wt(X)$, introduced in [6], has the property $wt(X)\leq t(X)$. It was shown in [4] that if $X$ is a homogeneous compactum then $|X|\leq 2^{wt(X)\pi\chi(X)}$. We introduce the almost tightness $at(X)$ with the property…
For a poset $P$, let $\sigma(P)$ and $\Gamma(P)$ respectively denote the lattice of its Scott open subsets and Scott closed subsets ordered by inclusion, and set $\Sigma P=(P,\sigma(P))$. In this paper, we discuss the lower Vietoris…
We further investigate the weak topology generated by the irreducible unitary representations of a group $G$. A deep result due to Ernest \cite{Ernest1971} and Hughes \cite{Hughes1973} asserts that every weakly compact subset of a locally…
Given a coarse space $(X, \mathcal{E})$, we consider linear orders on $X$ compatible with the coarse structure $\mathcal E$ and explore interplays between these orders and macro-uniform selectors of $(X, \mathcal{E})$.