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Intrinsic characterizations of C-realcompact spaces

Author: Rakesh Bharati
Deb Ray Atasi
Atasi Deb Ray
Acharyya Sudip Kumar
Sudip Kumar Acharyya
Bharati Rakesh
Publication venue
Publication date: 01/10/2021
Field of study

[EN] c-realcompact spaces are introduced by Karamzadeh and Keshtkar in Quaest. Math. 41, no. 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 properties. 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.University Grand Commission, New Delhi, research fellowship (F. No. 16-9 (June 2018)/2019 (NET/CSIR))Acharyya, SK.; Bharati, R.; Deb Ray, A. (2021). Intrinsic characterizations of C-realcompact spaces. Applied General Topology. 22(2):295-302. https://doi.org/10.4995/agt.2021.13696OJS295302222S. K. Acharyya and S. K. Ghosh, A note on functions in C(X) with support lying on an ideal of closed subsets of X, Topology Proc. 40 (2012), 297-301.S. K. Acharyya and S. K. Ghosh, Functions in C(X) with support lying on a class of subsets of X, Topology Proc. 35 (2010), 127-148.S. K. Acharyya, R. Bharati and A. Deb Ray, Rings and subrings of continuous functions with countable range, Queast. Math., to appear. https://doi.org/10.2989/16073606.2020.1752322F. Azarpanah, O. A. S. Karamzadeh, Z. Keshtkar and A. R. Olfati, On maximal ideals of

C_c(X)

and the uniformity of its localizations, Rocky Mountain J. Math. 48, no. 2 (2018), 345-384. https://doi.org/10.1216/RMJ-2018-48-2-345P. Bhattacherjee, M. L. Knox and W. W. Mcgovern, The classical ring of quotients of

C_c(X)

, Appl. Gen. Topol. 15, no. 2 (2014), 147-154. https://doi.org/10.4995/agt.2014.3181L. Gillman and M. Jerison, Rings of Continuous Functions, Van Nostrand Reinhold co., New York, 1960. https://doi.org/10.1007/978-1-4615-7819-2M. Ghadermazi, O. A. S. Karamzadeh and M. Namdari, On the functionally countable subalgebras of C(X), Rend. Sem. Mat. Univ. Padova. 129 (2013), 47-69. https://doi.org/10.4171/RSMUP/129-4O. A. S. Karamzadeh and Z. Keshtkar, On c-realcompact spaces, Queast. Math. 41, no. 8 (2018), 1135-1167. https://doi.org/10.2989/16073606.2018.1441919M. Mandelkar, Supports of continuous functions, Trans. Amer. Math. Soc. 156 (1971), 73-83. https://doi.org/10.1090/S0002-9947-1971-0275367-4R. M. Stephenson Jr, Initially k-compact and related spaces, in: Handbook of Set-Theoretic Topology, ed. Kenneth Kunen and Jerry E. Vaughan. Amsterdam, North-Holland, (1984) 603-632. https://doi.org/10.1016/B978-0-444-86580-9.50016-1A. Veisi,

e_c

-filters and

e_c

-ideals in the functionally countable subalgebra of

C^*(X)

, Appl. Gen. Topol. 20, no. 2 (2019), 395-405. https://doi.org/10.4995/agt.2019.1152

Poli[Papers] (Universitat Politècnica de València)

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

Author: Nandi Pratip
Ray Atasi Deb
Acharyya Sudip Kumar
Publication venue
Publication date: 19/06/2021
Field of study

In this article we introduce the zero-divisor graphs

\Gamma_\mathscr{P}(X)

and

\Gamma^\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

\Gamma_\mathscr{P}(X)

(respectively,

\Gamma^\mathscr{P}_\infty(X)

) becomes triangulated/ hypertriangulated. We realize that

\Gamma_\mathscr{P}(X)

(respectively,

\Gamma^\mathscr{P}_\infty(X)

) is a complemented graph if and only if the space of minimal prime ideals in

C_\mathscr{P}(X)

(respectively

\Gamma^\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}

X

and

Y

respectively that the rings

C_\mathscr{P}(X)

and

C_\mathscr{Q}(Y)

are isomorphic if and only if

\Gamma_\mathscr{P}(X)

and

\Gamma_\mathscr{Q}(Y)

are isomorphic

arXiv.org e-Print Archive

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

Author: Acharyya Soumyadip
Ray Atasi Deb
Acharyya Sudip Kumar
Bharati Rakesh
Publication venue
Publication date: 11/07/2022
Field of study

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

\mu

be a measure on (

X,\mathcal{A}

). In this paper, we generalize the so-called

m_{\mu}

and

U_{\mu}

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_{\mu_{I}}

and

U_{\mu_{I}}

topology, respectively, throughout the paper.

L_{I}^{\infty} \left(\mu\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}, \mu

)). Also let

I_{\mu} (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_{\mu} (X,\mathcal{A})

is an ideal in

\mathcal{M}(X,\mathcal{A})

containing

I

and contained in

L_{I}^{\infty} \left(\mu\right)

. It is also shown that

I_{\mu} (X,\mathcal{A})

and

L_{I}^{\infty} \left(\mu\right)

are the components of

0

in the spaces

m_{\mu_{I}}

and

U_{\mu_{I}}

, respectively. Additionally, we obtain a chain of necessary and sufficient conditions as to when these two topologies coincide

arXiv.org e-Print Archive

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

Author: Nandi Pratip
Ray Atasi Deb
Acharyya Sudip Kumar
Publication venue
Publication date: 03/06/2023
Field of study

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

\mathcal{M}(X,\mathcal{A})

and a measure

\mu:\mathcal{A}\to[0,\infty]

, we introduce the

U_\mu^I

-topology and the

m_\mu^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_\mu

-topology and the

m_\mu

-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_\mu^I

-topology and the

m_\mu^I

-topology coincide if and only if

X\setminus \bigcap Z[I]

is a

\mu

-bounded subset of

X

. The components of

0

\mathcal{M}(X,\mathcal{A})

in the

U_\mu^I

-topology and the

m_\mu^I

-topology are realized as

I\cap L^\infty(X,\mathcal{A},\mu)

and

I\cap L_\psi(X,\mathcal{A},\mu)

respectively. Here

L^\infty(X,\mathcal{A},\mu)

is the set of all functions in

\mathcal{M}(X,\mathcal{A})

which are essentially

\mu

-bounded over

X

and

L_\psi(X,\mathcal{A},\mu)=\{f\in \mathcal{M}(X,\mathcal{A}): ~\forall g\in\mathcal{M}(X,\mathcal{A}), f.g\in L^\infty(X,\mathcal{A},\mu)\}

. It is established that an ideal

I

\mathcal{M}(X,\mathcal{A})

is dense in the

U_\mu

-topology if and only if it is dense in the

m_\mu

-topology and this happens when and only when there exists

Z\in Z[I]

such that

\mu(Z)=0

. Furthermore, it is proved that

I

is closed in

\mathcal{M}(X,\mathcal{A})

in the

m_\mu

-topology if and only if it is a

Z_\mu

-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

arXiv.org e-Print Archive

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

Author: Nandi Pratip
Ray Atasi Deb
Acharyya Sudip Kumar
Bharati Rakesh
Publication venue
Publication date: 03/07/2023
Field of study

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]

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)

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

arXiv.org e-Print Archive

A generalization of m-topology and U-topology on rings of measurable functions

Author: Soumyadip Acharyya
A. Deb Ray
Rakesh Bharati
Acharyya Soumyadip
DEBRAY ATASI
Acharyya Sudip Kumar
Sudip Kumar Acharyya
Bharati Rakesh
Publication venue
Publication date: 01/04/2025
Field of study

In this paper, we generalize mμ and Uμ topologies on M ( X , A ) via an ideal I in the ring M ( X , A ) of all real-valued measurable functions. The collection LI∞ ( μ ) of all essentially I-bounded functions over the measure space (X,A,μ)$ and the ideal Iμ (X,A) ={ f ∈ M (X,A) : for every g ∈ M(X,A), fg is essentially I-bounded } are the components of 0 in the space mμI and UμI respectively. Additionally, we obtain a chain of necessary and sufficient conditions as to when these two topologies coincide. In particular, it is proved that they coincide if and only if the three sets M(X,A), LI∞ ( μ ) , and Iμ (X,A) coincide and this holds when and only when M(X,A), with either topology, is connected, which is further equivalent to each of these two topological spaces being a topological ring as well as a topological vector space. Furthermore, LI∞ ( μ ) is a complete pseudonormed linear space regardless of the choice of I. Finally, we examine when these later spaces are separable

Poli[Papers] (Universitat Politècnica de València)

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