arXiv subjects

Nur Alam

Publications and source records attributed to Nur Alam.

8 recordsLinked to original sources

arXiv2025

Certain results on selection principles associated with bornological structure in topological spaces

We study selection principles related to bornological covers in a topological space $X$ following the work of Aurichi et al., 2019, where selection principles have been investigated in the function space $C_\mathfrak{B}(X)$ endowed with the topology $\tau_\mathfrak{B}$ of uniform convergence on bornology $\mathfrak{B}$. We show equivalences among certain selection principles and present some game theoretic observations involving bornological covers. We investigate selection principles on the product space $X^n$ equipped with the product bornolgy $\mathfrak{B}^n$, $n\in \omega$. Considering the cardinal invariants such as the unbounding number ($\mathfrak{b}$), dominating numbers ($\mathfrak{d}$), pseudointersection numbers ($\mathfrak{p}$) etc., we establish connections between the cardinality of base of a bornology with certain selection principles. Finally, we investigate some variations of the tightness properties of $C_\mathfrak{B}(X)$ and present their characterizations in terms of selective bornological covering properties of $X$.

math.GN↗

arXiv2025

On a variation of selective separability using ideals

A space $X$ is H-separable (Bella et al., 2009) if for every sequence $(Y_n)$ of dense subspaces of $X$ there exists a sequence $(F_n)$ such that for each $n$ $F_n$ is a finite subset of $Y_n$ and every nonempty open set of $X$ intersects $F_n$ for all but finitely many $n$. In this paper, we introduce and study an ideal variant of H-separability, called $\mathcal{I}$-H-separability.

math.GN↗

arXiv2025

On a variation of selective separability: S-separability

A space $X$ is M-separable (selectively separable) (Scheepers, 1999; Bella et al., 2009) if for every sequence $(Y_n)$ of dense subspaces of $X$ there exists a sequence $(F_n)$ such that for each $n$ $F_n$ is a finite subset of $Y_n$ and $\cup_{n\in \mathbb{N}} F_n$ is dense in $X$. In this paper, we introduce and study a strengthening of M-separability situated between H- and M-separability, which we call S-separability: for every sequence $(Y_n)$ of dense subspaces of $X$ there exists a sequence $(F_n)$ such that for each $n$ $F_n$ is a finite subset of $Y_n$ and for each finite family $\mathcal F$ of nonempty open sets of $X$ some $n$ satisfies $U\cap F_n\neq\emptyset$ for all $U\in \mathcal F$.

math.GN↗

arXiv2023

On certain star versions of the Scheepers property

The star versions of the Scheepers property, namely star-Scheepers, strongly star-Scheepers and new star-Scheepers property have been introduced. We explore further ramifications concerning critical cardinalities. Quite a few interesting observations are obtained while dealing with the Isbell-Mrówka spaces, Niemytzki plane and Alexandroff duplicates. The properties like monotonically normal, locally countable cellularity (which is introduced here) play an important role in our investigation. We study games corresponding to the classical and star variants of the Scheepers property which have not been investigated in prior works. Some open problems are also posed.

math.GN↗

arXiv2023

On certain star-$K$ variant of a selection principle

This article deals with certain star variants of the Scheepers property. We introduce and study the star-$K$-Scheepers property and corresponding game. The relationships between the game corresponding to the star-$K$-Scheepers property and other variants of the Scheepers games are investigated. Furthermore, the relation among the winning strategies of the players in such games are represented in an implication diagram. We present certain observations in terms of Alster covers and the Alexandroff duplicates. In addition, we examine few preservation like properties in this context. Some open problems are also posed.

math.GN↗

arXiv2023

Further investigations on certain star selection principles

We consider certain star versions of the Menger, Hurewicz and Rothberger properties. Few important observations concerning these properties are presented, which have not been investigated in earlier works. A variety of investigations is performed using Alster covers and critical cardinalities $\mathfrak{d}$, $\mathfrak{b}$ and ${\sf cov}(\mathcal{M})$. Our study explores further ramifications on the extent and Alexandroff duplicate. In the process we present investigations on the star versions of the Rothberger property and compare with similar prior observations of the star versions of the Menger and Hurewicz properties. We sketch few tables that interpret (mainly preservation-kind of) properties of the star selection principles obtained so far. We also present implication diagrams to explicate the interplay between the star selection principles.

math.GN↗

arXiv2022

On certain localized version of uniform selection principles

We intend to localize the selection principles in uniform spaces (Kočinac, 2003) by introducing their local variations, namely locally $Υ$-bounded spaces (where $Υ$ is Menger, Hurewicz or Rothberger). It has been observed that the difference between uniform selection principles and the corresponding local correlatives as introduced here is reasonable enough to discuss about these new notions. Certain observations using the critical cardinals (on the uniform selection principles which have not studied before) as well as preservation like properties (on the local versions) are presented. The interrelationships between the notions considered in this paper are outlined into an implication diagram. Certain interactions between these local variations are also investigated. We present several examples to illustrate the distinguishable behaviour of the new notions.

math.GN↗

arXiv2022

On certain weaker forms of the Scheepers property

We introduce the weaker forms of the Scheepers property, namely almost Scheepers (${\sf aS}$), weakly Scheepers in the sense of Sakai (${\sf wS}$) and weakly Scheepers in the sense of Kočinac (${\sf wS_k}$). We explore many topological properties of the weaker forms of the Scheepers property and present few illustrative examples to make distinction between these spaces. Certain situations are considered when all the weaker forms are equivalent. We also make investigations on the weak variations as considered in this paper concerning cardinalities. In particular we observe that 1. If every finite power of a space $X$ is ${\sf aM}$ (respectively, ${\sf wM}$), then $X$ is ${\sf aS}$ (respectively, ${\sf wS}$). 2. Every almost Lindelöf space of cardinality less than $\mathfrak{d}$ is ${\sf aS}$. 3. Let $X$ be Lindelöf and $κ<\mathfrak d$. If $X$ is a union of $κ$ many ${\sf aH}$ (respectively, ${\sf wH}$, ${\sf wH_k}$) spaces, then $X$ is ${\sf aS}$ (respectively, ${\sf wS}$, ${\sf wS_k}$). 4. The Alexandroff duplicate $AD(X)$ of a space $X$ has the Scheepers property if and only if $AD(X)$ has the ${\sf wS_k}$ property. 5. If $AD(X)$ is ${\sf aS}$ (respectively, ${\sf wS}$), then $X$ is also ${\sf aS}$ (respectively, ${\sf wS}$). Besides, few observations on productively ${\sf aS}$, productively ${\sf wS}$ and productively ${\sf wS_k}$ spaces are presented. Some open problems are also given.

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