General Topology
The class of $L\Sigma(\leq\omega)$-spaces was introduced in 2006 by Kubi\'s, Okunev and Szeptycki as a natural refinement of the classical and important notion of Lindel\"of $\Sigma$-spaces. Compact $L\Sigma(\leq\omega)$-spaces were…
We give several characteristic properties of FAC spaces, namely topological spaces with no infinite discrete subspace. The first one was obtained in 2019 by the first author, and states that every closed set is a finite union of irreducible…
We present several characterizations of $\sigma$-compact Hattori spaces, and reject some possible characterization candidates of the spaces.
A central role in topological dimension theory is played by Lipscomb's space $J_{A}$ since it is a universal space for metric spaces of weight $|A|\geq \aleph _{0}$. On the one hand, Lipscomb's space is the attractor of a possibly infinite…
Motivated by the main results of the articles by Hattori and Bouziad, we seek to answer the following questions about Hattori spaces. Let A be a subset of the real line, then: Given a compact set $K$ in the Euclidean topology, under what…
Let R be a ring and X = SH(R)-{0} be the set of all non-zero strongly hollow ideals (briefly, sh-ideals) of R. We first study the concept SH-topology and investigate some of the basic properties of a topological space with this topology. It…
The Model Hypothesis (abbreviated $\mathsf{MH}$) and $\Delta$ are set-theoretic axioms introduced by J. Roitman in her work on the box product problem. Answering some questions of Roitman and Williams on these two principles, we show (1)…
Every directed set is Tukey equivalent to (a) the family of all compact subsets, ordered by inclusion, of a (locally compact) space, to (b) a neighborhood filter, ordered by reverse inclusion, of a point (of a compact space, and of a…
In this note, the Zorn lemma is extended to arbitrary binary relations and thus the Zorn lemma can do for optimization when the transitivity is broken. Zorn's extended lemma can be used to prove existence theorems of generalized solution…
The present paper refers to the notions of digital continuity, digital $k$-isomorphism, local $k$-isomorphism, radius $2$-local $k$-isomorphism, and digital $k$-homotopy motivated by the Khalimsky's version. We discuss something incorrectly…
One of the ways that connectedness has been studied through the history of topology is by using chains, the so called chain connectedness. Here we combine this notion together with continuity up to a covering to provide the inheritance of…
The paper refers to several concepts which are essential to studying digital objects from the viewpoint of digital topology: digital $k$-connectivity or digital $k$-adjacency, $C$-compatible and normal $k$-adjacency for a digital product.…
This paper aims to examine the version of the topological group structure in proximity and especially descriptive proximity spaces, that is, the concepts of proximal group and descriptive proximal group are introduced. In addition, the…
The main aim of this paper is to define a weakest topology $\sigma$ on a linear topological space $(E, \tau)$ such that each $\delta$-continuous functional on $(E, \tau)$ is $\delta$-continuous functional on $(E, \sigma)$ and to find out…
The notions of a local $(k_0,k_1)$-isomorphism and a weakly local $(k_0,k_1)$-isomorphism play crucial roles in developing a digital $(k_0,k_1)$-covering space and a pseudo-$(k_0,k_1)$-covering space, respectively. In relation to the study…
We will study two subclasses of the class of feebly compact spaces in the class of (para)topological groups, the compact-bounded and weakly compact-bounded spaces, both introduced by J. Angoa, Y. F. Ortiz-Castillo and A. Tamariz-Mascar\'ua…
In the following text we show that the Alexandroff space $X$ is uniformizable if and only if the collection of all smallest neighbourhoods is a partition of $X$. Moreover the Alexandroff space $X$ is uniformizable and functional Alexandroff…
We generalize the classic Vietoris endofunctor to the category of compact Hausdorff spaces and closed relations. The lift of a closed relation is done by generalizing the construction of the Egli-Milner order. We describe the dual…
We introduce a method that allows to turn topological questions about Hindman spaces into purely combinatorial questions about the Kat\v{e}tov order of ideals on $\mathbb{N}$. We also provide two applications of the method. (1) We…
In this paper we obtain new results regarding the chain conditions in the Pixley-Roy hyperspaces $\mathscr{F}[X]$. For example, if $c(X)$ and $R(X)$ denote the cellularity and weak separation number of $X$ (see Section~[4]) and we define…