arXiv subjects

A. Deb Ray

Publications and source records attributed to A. Deb Ray.

14 recordsLinked to original sources

arXiv2023

Lebesgue Property and $G$-completeness in Generalized Quasi-uniform Spaces

This paper extends the Lebesgue property and (weak) $G$-completeness to generalized quasi-uniform spaces. It investigates the connections between completeness, (weak) $G$-completeness, and the Lebesgue property of the product of generalized quasi-uniform spaces with those of the component spaces. It has been observed that the related behaviors differ for the Lebesgue property in contrast to completeness and (weak) $G$-completeness.

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arXiv2023

On a Generalization of Quasi-metric Space

We find an extension of the quasi-metric (to be called $g$-quasi metric) such that the induced generalized topology may fail to form a topology. We show that $g$-quasi metrizability is a $g$-topologically invariant property of generalized topological spaces. Extending metric product and uniform continuity for $g$-quasi metric spaces, we note that a $g$-quasi metric may fail to be uniformly continuous in the extended sense unlike usual metric. Finally, we extend the study of completeness, Lebesgue property and weak $G$-completeness for $g$-quasi metric spaces.

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arXiv2022

Zero-divisor graph and comaximal graph of rings of continuous functions with countable range

In this paper, two outwardly different graphs, namely, the zero divisor graph $\Gamma(C_c(X))$ and the comaximal graph $\Gamma_2^{'}(C_c(X))$ of the ring $C_c(X)$ of all real-valued continuous functions having countable range, defined on any Hausdorff zero dimensional space $X$, are investigated. It is observed that these two graphs exhibit resemblance, so far as the diameters, girths, connectedness, triangulatedness or hypertriangulatedness. are concerned. However, the study reveals that the zero divisor graph $\Gamma(A_c(X))$ of an intermediate ring $A_c(X)$ of $C_c(X)$ is complemented if and only if the space of all minimal prime ideals of $A_c(X)$ is compact. Moreover, $\Gamma(C_c(X))$ is complemented when and only when its subgraph $\Gamma(A_c(X))$ is complemented. On the other hand, the comaximal graph of $C_c(X)$ is complemented if and only if the comaximal graph of its over-ring $C(X)$ is complemented and the latter graph is known to be complemented if and only if $X$ is a $P$-space. Indeed, for a large class of spaces (i.e., for perfectly normal, strongly zero dimensional spaces which are not P-spaces), $\Gamma(C_c(X))$ and $\Gamma_2^{'}(C_c(X))$ are seen to be non-isomorphic. Defining appropriately the quotient of a graph, it is utilised to establish that for a discrete space $X$, $\Gamma(C_c(X))$ (= $\Gamma(C(X))$) and $\Gamma_2^{'}(C_c(X))$ (= $\Gamma_2^{'}(C(X))$) are isomorphic, if $X$ is atmost countable. Under the assumption of continuum hypothesis, the converse of this result is also shown to be true.

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arXiv2021

Common Fixed Point Theorems on Complete and Weak $G$-Complete Fuzzy Metric Spaces

Motivated by Gopal and Vetro [Iranian Journal of Fuzzy Systems, 11(3), 95-107], we introduce a symmetric pair of $\beta$-admissible mappings and obtain common fixed point theorems for such a pair in complete and weak $G$-complete fuzzy metric spaces. In particular, we rectified, generalize and improve the common fixed point theorem obtained by Turkoglu and Sangurlu [Journal of Intelligent & Fuzzy Systems, 26(1), 137-142] for two fuzzy $\psi$-contractive mappings. We include non-trivial examples to exhibit the generality and demonstrate our results.

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arXiv2021

Real compactness via real maximal ideals of $B_1(X)$

In this paper, constructing a class of ideals of $B_1(X)$ from proper ideals of $C(X)$ a one-one correspondence between the class of real maximal ideals of $C(X)$ and those of $B_1(X)$ is established. The collection of all real maximal ideals of $B_1(X)$ with hull-kernel topology is proved to be homeomorphic to the space of real maximal ideals of $C(X)$ endowed with a topology finer than the subspace topology induced from its structure space. It is also proved that a Tychonoff space is real compact if and only if every real maximal ideal of $B_1(X)$ is fixed. As a consequence, within the class of real compact $T_4$ spaces whose points are $G_\delta$, $B_1(X) = {B_1}^*(X)$ if and only if $X$ is finite.

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arXiv2020

More on the rings $B_1(X)$ and $B_1^*(X)$

This paper focuses mainly on the ring of all bounded Baire one functions on a topological space. The uniform norm topology arises from the $\sup$-norm defined on the collection $B_1^*(X)$ of all bounded Baire one functions. With respect to this topology, $B_1^*(X)$ is a topological ring. It is proved that under uniform norm topology, the set of all units forms an open set and as a consequence of it, every maximal ideal of $B_1^*(X)$ is closed in $B_1^*(X)$ with uniform norm topology. Since the natural extension of uniform norm topology on $B_1(X)$, when $B_1^*(X) \neq B_1(X)$, does not show up these features, a topology called $m_B$-topology is defined on $B_1(X)$ suitably to achieve these results on $B_1(X)$. It is proved that the relative $m_B$ topology coincides with the uniform norm topology on $B_1^*(X)$ if and only if $B_1(X) = B_1^*(X)$. Moreover, $B_1(X)$ with $m_B$-topology is 1st countable if and only if $B_1(X) = B_1^*(X)$. \\ The last part of the paper establishes a correspondence between the ideals of $B_1^*(X)$ and a special class of $Z_B$-filters, called $e_B$-filters on a normal topological space $X$. It is also observed that for normal spaces, the cardinality of the collection of all maximal ideals of $B_1(X)$ and those of $B_1^*(X)$ are the same.

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arXiv2020

On Weak $G$-Completeness for Fuzzy Metric Spaces

In this paper, we provide equivalent characterizations of weak $G$-complete fuzzy metric spaces. Since such spaces are complete, we also characterize fuzzy metric spaces that have weak $G$-complete fuzzy metric completions. Moreover we establish analogous results for classical metric spaces.

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arXiv2020

On Structure space of the ring $B_1(X)$

In this article, we continue our study of the ring of Baire one functions on a topological space $(X,\tau)$, denoted by $B_1(X)$ and extend the well known M. H. Stones's theorem from $C(X)$ to $B_1(X)$. Introducing the structure space of $B_1(X)$, an analogue of Gelfand Kolmogoroff theorem is established. It is observed that $(X,\tau)$ may not be embedded inside the structure space of $B_1(X)$. This observation inspired us to introduce a weaker form of embedding and show that in case $X$ is a $T_4$ space, $X$ is weakly embedded as a dense subspace, in the structure space of $B_1(X)$. It is further established that the ring $B_1^{*}(X)$ of all bounded Baire one functions is a C-type ring and also, the structure space of $B_1^{*}(X)$ is homeomorphic to the structure space of $B_1(X)$. Introducing a finer topology $\sigma$ than the original $T_4$ topology $\tau$ on $X$, it is proved that $B_1(X)$ contains free (maximal) ideals if $\sigma$ is strictly finer than $\tau$. It is also proved that $\tau = \sigma$ if and only if $B_1(X) = C(X)$. Moreover, in the class of all perfectly normal $T_1$ spaces, $B_1(X) = C(X)$ is equivalent to the discreteness of the space $X$.

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arXiv2020

Some Properties of Lebesgue Fuzzy Metric Spaces

In this paper, we establish a sequential characterisation of Lebesgue fuzzy metric and explore the relationship between Lebesgue, weak G-complete and compact fuzzy metric spaces. We also discuss the Lebesgue property of several well-known fuzzy metric spaces.

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arXiv2019

Rings and subrings of continuous functions with countable range

Intermediate rings of real valued continuous functions with countable range on a Hausdorff zero-dimensional space $X$ are introduced in this article. Let $\Sigma_c(X)$ be the family of all such intermediate rings $A_c(X)$'s which lie between $C_c^*(X)$ and $C_c(X)$. It is shown that the structure space of each $A_c(X)$ is $\beta_0X$, the Banaschewski compactification of $X$. $X$ is shown to be a $P$-space if and only if each ideal in $C_c(X)$ is closed in the $m_c$-topology on it. Furthermore $X$ is realized to be an almost $P$-space when and only when each maximal ideal/ $z$-ideal in $C_c(X)$ becomes a $z^0$-ideal. Incidentally within the family of almost $P$-spaces, $C_c(X)$ is characterized among all the members of $\Sigma_c(X)$ by virtue of either of these two properties. Equivalent descriptions of pseudocompact condition on $X$ are given via $U_c$-topology, $m_c$-topology and norm on $C_c(X)$. The article ends with a result which essentially says that $z^0$-ideals in a typical $A_c(X)$ $\in$ $\Sigma_c(X)$ are precisely the contraction of $z^0$-ideals in $C_c(X)$.

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arXiv2019

A $T_0$-Compactification Of A Tychonoff Space Using The Rings Of Baire One Functions

In this article, we continue our study of Baire one functions on a topological space $X$, denoted by $B_1(X)$ and extend the well known M. H. Stones's theorem from $C(X)$ to $B_1(X)$. Introducing the structure space of $B_1(X)$, it is observed that $X$ may not be embedded inside this structure space. This observation inspired us to build a space $\mathcal{M}(B_1(X))/\sim$, from the structure space of $B_1(X)$ and to show that $X$ is densely embedded in $\mathcal{M}(B_1(X))/\sim$. It is further established that it is a $T_0$-compactification of $X$. Such compactification of $X$ possesses the extension property for continuous functions, though it lacks Hausdorffness in general. Therefore, it is natural to search for condition(s) under which it becomes Hausdorff. In the last section, a set of necessary and sufficient conditions for such compactification to become a Stone-Ceck compatification, is finally arrived at.

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arXiv2019

Ideals in $B_1(X)$ and residue class rings of $B_1(X)$ modulo an ideal

This paper explores the duality between ideals of the ring $B_1(X)$ of all real valued Baire one functions on a topological space $X$ and typical families of zero sets, called $Z_B$-filters, on $X$. As a natural outcome of this study, it is observed that $B_1(X)$ is a Gelfand ring but non-Noetherian in general. Introducing fixed and free maximal ideals in the context of $B_1(X)$, complete descriptions of the fixed maximal ideals of both $B_1(X)$ and $B_1^*(X)$ are obtained. Though free maximal ideals of $B_1(X)$ and those of $B_1^*(X)$ do not show any relationship in general, their counterparts, i.e., the fixed maximal ideals obey natural relations. It is proved here that for a perfectly normal $T_1$ space $X$, free maximal ideals of $B_1(X)$ are determined by a typical class of Baire one functions. In the concluding part of this paper, we study residue class ring of $B_1(X)$ modulo an ideal, with special emphasize on real and hyper real maximal ideals of $B_1(X)$.

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arXiv2018

On Lebesgue Property for Fuzzy Metric Spaces

We provide several characterizations of the Lebesgue property for fuzzy metric spaces. It is known that a fuzzy metric space is Lebesgue if and only if every real-valued continuous function is uniformly continuous. Here we show that it suffices to examine uniform continuity of bounded real-valued continuous functions for characterizing Lebesgue property in fuzzy setting.

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arXiv2018

On Rings of Baire one functions

This paper introduces the ring of all real valued Baire one functions, denoted by $B_1(X)$ and also the ring of all real valued bounded Baire one functions, denoted by $B_1^*(X)$. Though the resemblance between $C(X)$ and $B_1(X)$ is the focal theme of this paper, it is observed that unlike $C(X)$ and $C^*(X)$ (real valued bounded continuous functions), $B_1^*(X)$ is a proper subclass of $B_1(X)$ in almost every non-trivial situation. Introducing $B_1$-embedding and $B_1^*$-embedding, several analogous results, especially, an analogue of Urysohn's extension theorem is established.

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