arXiv subjects

Sudip Kumar Acharyya

Publications and source records attributed to Sudip Kumar Acharyya.

At least 19 recordsLinked to original sources

arXiv2026

A topological view of algebraic structures in $C_p(X)$

This manuscript focuses on the topological behaviour of algebraic structures in $C_p(X)$. Closure of ideals in $C_p(X)$ is determined and are used to characterise various topological properties of the underlying Tychonoff space $X$. These include compact space, locally compact space, locally pseudocompact space, $P$-space, almost $P$-space and normal space. Moreover, it has been established that a space $X$ is totally separated space if and only if the set of all units in $C(X)$ is dense in $C_p(X)$. A subring of $C(X)$ is found to be dense in $C_p(X)$ when and only when it separates points.

math.GN↗

arXiv2026

A special kind of topology on $C(X)$ lying between the point-open topology and the topology of uniform convergence

Let $X$ be a topological space. For each transfinite cardinal number $\aleph_α$, we define a topology $C_{\aleph_α}(X)$ on the ring $C(X)$. With $\aleph_α=\aleph_0$, $C_{\aleph_α}(X)$ reduces to the space $C_p(X)$. We prove that for $\aleph_α\geq \aleph_1$, $C_{\aleph_α}(X)$ is pathwise connected if and only if it is connected if and only if $X$ is pseudocompact. Here, we define $\aleph_α$-separable space and furthermore, we show that $X$ is $\aleph_α$-separable when and only when $C_{\aleph_α}(X)$ is metrizable when and only when it is a sequential space. Later we introduce two new cardinal functions, namely $cc_{\aleph_α}(X)$ and $ac_{\aleph_α}(X)$ which turned out to be the character and pseudocharacter of $C_{\aleph_α}(X)$ respectively. At the end of this article we show that a number cardinal functions associated with the space $C_{\aleph_α}(X)$ are equal.

math.GN↗

arXiv2026

A number of properties enjoyed by two specially constructed topologies on $C(X)$

If $I$ is an ideal in the ring $C(X)$ of all real valued continuous functions defined over a Tychonoff space $X$, then $X$ is called $I$-$pseudocompact$ if the set $X\setminus \bigcap Z[I]$ is a bounded subset of $X$. Corresponding to $I$, the $m^I$-topology and $u^I$-topology on $C(X)$, generalizing the well-known $m$-topology and $u$-topology in $C(X)$ respectively are already there in the literature. It is proved amongst others that the $m^I$-topology is first countable if and only if the $u^I$-topology= $m^I$-topology on $C(X)$ if and only if $X$ is $I$-$pseudocompact$. A special case of this result on choosing $I=C(X)$ reads: the $u$-topology and $m$-topology on $C(X)$ coincide if and only if $X$ is pseudocompact. It is established that the $m^I$-topology on $C(X)$ is second countable if and only if it is $\aleph_0$-$bounded$ if and only if $X$ is compact, metrizable and $I=C(X)$. Furthermore it is realized that the $m^I$ topology on $C(X)$ is hemicompact if and only if it is $σ$-compact if and only if this topology is $H$-$bounded$ if and only if $X$ is finite and $I=C(X)$.

math.GN↗

arXiv2025

Measure finite topology on the ring of measurable functions

Let $\mathcal{M}(X,\mathcal{A},μ)$ be the ring of all real-valued measurable functions constructed over a measure space $(X,\mathcal{A},μ)$. A topology on $\mathcal{M}(X,\mathcal{A},μ)$, called the {$F_μ$-topology} weaker than the { $U_μ$-topology} is introduced. It is realized that the {component}, the {quasi component} and the {path component }in this {$F_μ$-topology} are identical. It turns out that the {$F_μ$-topology} on $\mathcal{M}(X,\mathcal{A},μ)$ becomes {connected} if and only if it is {path connected} if and only if $μ$ is an {atomic measure} of a special type. It is also proved that the {$F_μ$-topology} is {first countable} when and only when $μ$ is a {hemifinite measure.} Finally, it is shown that the {second countability} of the {$F_μ$-topology} is equivalent to the {hemifiniteness} of the measure $μ$ together with the {countable chain condition} of the {$F_μ$-topology}.

math.GN↗

arXiv2024

Structure spaces and allied problems on a class of rings of measurable functions

A ring $S(X,\mathcal{A})$ of real valued $\mathcal{A}$-measurable functions defined over a measurable space $(X,\mathcal{A})$ is called a $χ$-ring if for each $E\in \mathcal{A} $, the characteristic function $χ_{E}\in S(X,\mathcal{A})$. The set $\mathcal{U}_X$ of all $\mathcal{A}$-ultrafilters on $X$ with the Stone topology $τ$ is seen to be homeomorphic to an appropriate quotient space of the set $\mathcal{M}_X$ of all maximal ideals in $S(X,\mathcal{A})$ equipped with the hull-kernel topology $τ_S$. It is realized that $(\mathcal{U}_X,τ)$ is homeomorphic to $(\mathcal{M}_S,τ_S)$ if and only if $S(X,\mathcal{A})$ is a Gelfand ring. It is further observed that $S(X,\mathcal{A})$ is a Von-Neumann regular ring if and only if each ideal in this ring is a $\mathcal{Z}_S$-ideal and $S(X,\mathcal{A})$ is Gelfand when and only when every maximal ideal in it is a $\mathcal{Z}_S$-ideal. A pair of topologies $u_μ$-topology and $m_μ$-topology, are introduced on the set $S(X,\mathcal{A})$ and a few properties are studied.

math.GN↗

arXiv2023

Some familiar graphs on the rings of measurable functions

In this paper, replacing `equality' by 'equality almost everywhere' we modify several terms associated with the ring of measurable functions defined on a measure space $(X, \mathcal{A}, μ)$ and thereby study the graph theoretic features of the modified comaximal graph, annihilator graph and the weakly zero-divisor graph of the said ring. The study reveals a structural analogy between the modified versions of the comaximal and the zero-divisor graphs, which prompted us to investigate whether these two graphs are isomorphic. Introducing a quotient-like concept, we find certain subgraphs of the comaximal graph and the zero-divisor graph of $\mathcal{M}(X, \mathcal{A})$ and show that these two subgraphs are always isomorphic. Choosing $μ$ as a counting measure, we prove that even if these two induced graphs are isomorphic, the parent graphs may not be so. However, in case of Lebesgue measure space on $\mathbb{R}$, we establish that the comaximal and the zero-divisor graphs are isomorphic. Observing that both of the comaximal and the zero-divisor graphs of the ring $\mathcal{M}(X, \mathcal{A})$ are subgraphs of the annihilator graph of the said ring, we find equivalent conditions for their equalities in terms of the partitioning of $X$ into two atoms. Moreover, the non-atomicity of the underlying measure space $X$ is characterized through graph theoretic phenomena of the comaximal and the annihilator graph of $\mathcal{M}(X, \mathcal{A})$.

math.GM↗

arXiv2023

More on generalizations of topology of uniform convergence and $m$-topology on $C(X)$

This paper conglomerates our findings on the space $C(X)$ of all real valued continuous functions, under different generalizations of the topology of uniform convergence and the $m$-topology. The paper begins with answering all the questions which were left open in our previous paper on the classifications of $Z$-ideals of $C(X)$ induced by the $U_I$ and the $m_I$-topologies on $C(X)$. Motivated by the definition of $m^I$-topology, another generalization of the topology of uniform convergence, called $U^I$-topology, is introduced here. Among several other results, it is established that for a convex ideal $I$, a necessary and sufficient condition for $U^I$-topology to coincide with $m^I$-topology is the boundedness of $X\setminus\bigcap Z[I]$ in $X$. As opposed to the case of the $U_I$-topologies (and $m_I$-topologies), it is proved that each $U^I$-topology (respectively, $m^I$-topology) on $C(X)$ is uniquely determined by the ideal $I$. In the last section, the denseness of the set of units of $C(X)$ in $C_U(X)$ (= $C(X)$ with the topology of uniform convergence) is shown to be equivalent to the strong zero dimensionality of the space $X$. Also, the space $X$ is a weakly P-space if and only if the set of zero divisors (including 0) in $C(X)$ is closed in $C_U(X)$. Computing the closure of $C_\mathscr{P}(X)$ (=$\{f\in C(X):\text{the support of }f\in\mathscr{P}\}$ where $\mathscr{P}$ denotes the ideal of closed sets in $X$) in $C_U(X)$ and $C_m(X)$ (= $C(X)$ with the $m$-topology), the results $cl_UC_\mathscr{P}(X) = C_\infty^\mathscr{P}(X)$ ($=\{f\in C(X):\forall n\in\mathbb{N}, \{x\in X:|f(x)|\geq\frac{1}{n}\}\in\mathscr{P}\}$) and $cl_mC_\mathscr{P}(X)=\{f\in C(X):f.g\in C^\mathscr{P}_\infty(X)\text{ for each }g\in C(X)\}$ are achieved.

math.GN↗

arXiv2023

$U$-topology and $m$-topology on the ring of Measurable Functions, generalized and revisited

Let $\mathcal{M}(X,\mathcal{A})$ be the ring of all real valued measurable functions defined over the measurable space $(X,\mathcal{A})$. Given an ideal $I$ in $\mathcal{M}(X,\mathcal{A})$ and a measure $μ:\mathcal{A}\to[0,\infty]$, we introduce the $U_μ^I$-topology and the $m_μ^I$-topology on $\mathcal{M}(X,\mathcal{A})$ as generalized versions of the topology of uniform convergence or the $U$-topology and the $m$-topology on $\mathcal{M}(X,\mathcal{A})$ respectively. With $I=\mathcal{M}(X,\mathcal{A})$, these two topologies reduce to the $U_μ$-topology and the $m_μ$-topology on $\mathcal{M}(X,\mathcal{A})$ respectively, already considered before. If $I$ is a countably generated ideal in $\mathcal{M}(X,\mathcal{A})$, then the $U_μ^I$-topology and the $m_μ^I$-topology coincide if and only if $X\setminus \bigcap Z[I]$ is a $μ$-bounded subset of $X$. The components of $0$ in $\mathcal{M}(X,\mathcal{A})$ in the $U_μ^I$-topology and the $m_μ^I$-topology are realized as $I\cap L^\infty(X,\mathcal{A},μ)$ and $I\cap L_ψ(X,\mathcal{A},μ)$ respectively. Here $L^\infty(X,\mathcal{A},μ)$ is the set of all functions in $\mathcal{M}(X,\mathcal{A})$ which are essentially $μ$-bounded over $X$ and $L_ψ(X,\mathcal{A},μ)=\{f\in \mathcal{M}(X,\mathcal{A}): ~\forall g\in\mathcal{M}(X,\mathcal{A}), f.g\in L^\infty(X,\mathcal{A},μ)\}$. It is established that an ideal $I$ in $\mathcal{M}(X,\mathcal{A})$ is dense in the $U_μ$-topology if and only if it is dense in the $m_μ$-topology and this happens when and only when there exists $Z\in Z[I]$ such that $μ(Z)=0$. Furthermore, it is proved that $I$ is closed in $\mathcal{M}(X,\mathcal{A})$ in the $m_μ$-topology if and only if it is a $Z_μ$-ideal in the sense that if $f\equiv g$ almost everywhere on $X$ with $f\in I$ and $g\in\mathcal{M}(X,\mathcal{A})$, then $g\in I$.

math.GN↗

arXiv2023

Rings of functions whose closure of discontinuity set is in an ideal of closed sets

Let $\mathcal{P}$ be an ideal of closed subsets of a topological space $X$. Consider the ring, $C(X)_\mathcal{P}$ of real valued functions on $X$ whose closure of discontinuity set is a member of $\mathcal{P}$. We investigate the ring properties of $C(X)_\mathcal{P}$ for different choices of $\mathcal{P}$, such as the $\aleph_0$-self injectivity and regularity of the ring, if and when the ring is Artinian and/or Noetherian. The concept of $\mathcal{F}P$-space was introduced by Z. Gharabaghi, M. Ghirati and A. Taherifar in 2018 in a paper published in Houston Journal of Mathematics. In this paper, they established a result stating that every $P$-space is a $\mathcal{F}P$-space. We furnish that this theorem might fail if $X$ is not Tychonoff and we provide a suitable counter example to prove our assertion.

math.GN↗

arXiv2022

A Generalization of $ m $-topology and $ U $-topology on rings of measurable functions

For a measurable space ($X,\mathcal{A}$), let $\mathcal{M}(X,\mathcal{A})$ be the corresponding ring of all real valued measurable functions and let $μ$ be a measure on ($X,\mathcal{A}$). In this paper, we generalize the so-called $m_μ$ and $U_μ$ topologies on $\mathcal{M}(X,\mathcal{A})$ via an ideal $I$ in the ring $\mathcal{M}(X,\mathcal{A})$. The generalized versions will be referred to as the $m_{μ_{I}}$ and $U_{μ_{I}}$ topology, respectively, throughout the paper. $L_{I}^{\infty} \left(μ\right)$ stands for the subring of $\mathcal{M}(X,\mathcal{A})$ consisting of all functions that are essentially $I$-bounded (over the measure space ($X,\mathcal{A}, μ$)). Also let $I_μ (X,\mathcal{A}) = \big \{ f \in \mathcal{M}(X,\mathcal{A}) : \, \text{for every} \, g \in \mathcal{M}(X,\mathcal{A}), fg \, \, \text{is essentially} \, I$-$\text{bounded} \big \}$. Then $I_μ (X,\mathcal{A})$ is an ideal in $\mathcal{M}(X,\mathcal{A})$ containing $I$ and contained in $L_{I}^{\infty} \left(μ\right)$. It is also shown that $I_μ (X,\mathcal{A})$ and $L_{I}^{\infty} \left(μ\right)$ are the components of $0$ in the spaces $m_{μ_{I}}$ and $U_{μ_{I}}$, respectively. Additionally, we obtain a chain of necessary and sufficient conditions as to when these two topologies coincide.

math.FA↗

arXiv2022

Annihilator graph of the ring $C_\mathscr{P}(X)$

In this article, we introduce the annihilator graph of the ring $C_\mathscr{P}(X)$, denoted by $AG(C_\mathscr{P}(X))$ and observe the effect of the underlying Tychonoff space $X$ on various graph properties of $AG(C_\mathscr{P}(X))$. $AG(C_\mathscr{P}(X))$, in general, lies between the zero divisor graph and weakly zero divisor graph of $C_\mathscr{P}(X)$ and it is proved that these three graphs coincide if and only if the cardinality of the set of all $\mathscr{P}$-points, $X_\mathscr{P}$ is $\leq 2$. Identifying a suitable induced subgraph of $AG(C_\mathscr{P}(X))$, called $G(C_\mathscr{P}(X))$, we establish that both $AG(C_\mathscr{P}(X))$ and $G(C_\mathscr{P}(X))$ share similar graph theoretic properties and have the same values for the parameters, e.g., diameter, eccentricity, girth, radius, chromatic number and clique number. By choosing the ring $C_\mathscr{P}(X)$ where $\mathscr{P}$ is the ideal of all finite subsets of $X$ such that $X_\mathscr{P}$ is finite, we formulate an algorithm for coloring the vertices of $G(C_\mathscr{P}(X))$ and thereby get the chromatic number of $AG(C_\mathscr{P}(X))$. This exhibits an instance of coloring infinite graphs by just a finite number of colors. We show that any graph isomorphism $ψ: AG(C_\mathscr{P}(X)) \rightarrow AG(C_\mathscr{Q}(Y))$ maps $G(C_\mathscr{P}(X))$ isomorphically onto $G(C_\mathscr{Q}(Y))$ as a graph and a graph isomorphism $ϕ: G(C_\mathscr{P}(X)) \rightarrow G(C_\mathscr{Q}(Y))$ can be extended to a graph isomorphism $ψ: AG(C_\mathscr{P}(X)) \rightarrow AG(C_\mathscr{Q}(Y))$ under a mild restriction on the function $ϕ$. Finally, we show that atleast for the rings $C_\mathscr{P}(X)$ with finitely many $\mathscr{P}$-points, so far as the graph properties are concerned, the induced subgraph $G(C_\mathscr{P}(X))$ is a good substitute for $AG(C_\mathscr{P}(X)$.

math.GM↗

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 $Γ(C_c(X))$ and the comaximal graph $Γ_2^{'}(C_c(X))$ 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 $Γ(A_c(X))$ 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, $Γ(C_c(X))$ is complemented when and only when its subgraph $Γ(A_c(X))$ 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), $Γ(C_c(X))$ and $Γ_2^{'}(C_c(X))$ are seen to be non-isomorphic. Defining appropriately the quotient of a graph, it is utilised to establish that for a discrete space $X$, $Γ(C_c(X))$ (= $Γ(C(X))$) and $Γ_2^{'}(C_c(X))$ (= $Γ_2^{'}(C(X))$) are isomorphic, if $X$ is atmost countable. Under the assumption of continuum hypothesis, the converse of this result is also shown to be true.

math.GN↗

arXiv2021

Zero-divisor graph of the rings $C_\mathscr{P}(X)$ and $C^\mathscr{P}_\infty(X)$

In this article we introduce the zero-divisor graphs $Γ_\mathscr{P}(X)$ and $Γ^\mathscr{P}_\infty(X)$ of the two rings $C_\mathscr{P}(X)$ and $C^\mathscr{P}_\infty(X)$; here $\mathscr{P}$ is an ideal of closed sets in $X$ and $C_\mathscr{P}(X)$ is the aggregate of those functions in $C(X)$, whose support lie on $\mathscr{P}$. $C^\mathscr{P}_\infty(X)$ is the $\mathscr{P}$ analogue of the ring $C_\infty (X)$. We find out conditions on the topology on $X$, under-which $Γ_\mathscr{P}(X)$ (respectively, $Γ^\mathscr{P}_\infty(X)$) becomes triangulated/ hypertriangulated. We realize that $Γ_\mathscr{P}(X)$ (respectively, $Γ^\mathscr{P}_\infty(X)$) is a complemented graph if and only if the space of minimal prime ideals in $C_\mathscr{P}(X)$ (respectively $Γ^\mathscr{P}_\infty(X)$) is compact. This places a special case of this result with the choice $\mathscr{P}\equiv$ the ideals of closed sets in $X$, obtained by Azarpanah and Motamedi in \cite{Azarpanah} on a wider setting. We also give an example of a non-locally finite graph having finite chromatic number. Finally it is established with some special choices of the ideals $\mathscr{P}$ and $\mathscr{Q}$ on $X$ and $Y$ respectively that the rings $C_\mathscr{P}(X)$ and $C_\mathscr{Q}(Y)$ are isomorphic if and only if $Γ_\mathscr{P}(X)$ and $Γ_\mathscr{Q}(Y)$ are isomorphic.

math.AC↗

arXiv2021

Abundance of Isomorphic and non isomorphic intermediate rings

It is well known that for a non pseudocompact space X, the family (X) of all intermediate subrings of C(X) which contain bounded real valued continuous functions contains at least 2c many distinct rings. We show that if in addition X is first countable and real compact, then there are at least 2c many rings in (X), no two of which are pairwise isomorphic.

math.GN↗

arXiv2020

Ordered field valued continuous functions with countable range

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 an uncountable field or countable subfield of $\mathbb{R}$, then the structure space of $C_c(X,F)$ is $β_0X$, the Banaschewski Compactification of $X$. The ideals $\{O^{p,F}_c:p\in β_0X\}$ in $C_c(X,F)$ are introduced as modified countable analogue of the ideals $\{O^p:p\inβX\}$ in $C(X)$. It is realized that $C_c(X,F)\cap C_K(X,F)=\bigcap_{p\inβ_0X\texttt{\textbackslash}X} O^{p,F}_c$, this may be called a countable analogue of the well-known formula $C_K(X)=\bigcap_{p\inβX\texttt{\textbackslash}X}O^p$ in $C(X)$. Furthermore, it is shown that the hypothesis $C_c(X,F)$ is a Von-Neumann regular ring is equivalent to amongst others the condition that $X$ is a $P-$space.

math.GN↗

arXiv2020

Intrinsic characterizations of $C$-realcompact spaces

$c$-realcompact spaces are introduced by Karamzadeh and Keshtkar in Quaest. Math. 41(8), 2018, 1135-1167. We offer a characterization of these spaces $X$ via $c$-stable family of closed sets in $X$ by showing that $X$ is $c$-realcompact if and only if each $c$-stable family of closed sets in $X$ with finite intersection property has nonempty intersection. This last condition which makes sense for an arbitrary topological space can be taken as an alternative definition of a $c$-realcompact space. We show that each topological space can be extended as a dense subspace to a $c$-realcompact space with some desired extension property. An allied class of spaces viz $CP$-compact spaces akin to that of $c$-realcompact spaces are introduced. The paper ends after examining how far a known class of $c$-realcompact spaces could be realized as $CP$-compact for appropriately chosen ideal $P$ of closed sets in $X$.

math.GN↗

arXiv2020

Intermediate rings of complex-valued continuous functions

Let $Σ(X,\mathbb{C})$ denote the collection of all the rings between $C^*(X,\mathbb{C})$ and $C(X,\mathbb{C})$. We show that there is a natural correlation between the absolutely convex ideals/ prime ideals/maximal ideals/$z$-ideals/$z^\circ$-ideals in the rings $P(X,\mathbb{C})$ in $Σ(X,\mathbb{C})$ and in their real-valued counterparts $P(X,\mathbb{C})\cap C(X)$. It is shown that the structure space of any such $P(X,\mathbb{C})$ is $βX$. We show that for any maximal ideal $M$ in $C(X,\mathbb{C}), C(X,\mathbb{C})/M$ is an algebraically closed field. We give a necessary and sufficient condition for the ideal $C_{\mathcal{P}}(X,\mathbb{C})$ of $C(X,\mathbb{C})$ to be a prime ideal, and we examine a few special cases thereafter.

math.GN↗

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 $Σ_c(X)$ 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 $β_0X$, 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 $Σ_c(X)$ 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 $A_c(X)$ $\in$ $Σ_c(X)$ are precisely the contraction of $z^0$-ideals in $C_c(X)$.

math.GN↗