arXiv subjects

Soumajit Dey

Publications and source records attributed to Soumajit Dey.

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