On some topology generated by $\mathcal{I}$-density function

Indrajit Debnath; Amar Kumar Banerjee

Home/Archive/arXiv

arXiv · 2307.10777

On some topology generated by $\mathcal{I}$-density function

Indrajit Debnath·Amar Kumar Banerjee

View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper we have studied on $\mathcal{I}$-density function using the notion of $\mathcal{I}$-density, introduced by Banerjee and Debnath \cite{banerjee 4} where $\mathcal{I}$ is an ideal of subsets of the set of natural numbers. We have explored certain properties of $\mathcal{I}$-density function and induced a topology using this function in the space of reals namely $\mathcal{I}$-density topology and we have given a characterization of the Lebesgue measurable subsets of reals in terms of Borel sets in $\mathcal{I}$-density topology.

Explore related subjects

math.GN

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Indrajit Debnath, Amar Kumar Banerjee. 2023-07-20. On some topology generated by $\mathcal{I}$-density function. https://arxiv.org/abs/2307.10777

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Sobriety of Scott topologies under countability conditions

In this paper, we focus on the sobriety of the Scott topology in countable case. Specifically, we show that: (1) every meet-continuous core-compact countable dcpo is sober with respect to the Scott topology; (2)every countable locally compact dcpo is sober endowed with the Scott topology; (3) the lattice of all open sets for the rational numbers space Q equipped with the Scott topology is not sober.

math.GN↗

arXiv2026

Scott--Isbell Coincidence for Continuous Dcpos beyond Bicompleteness

Lawson and Mislove posed the following problem in 1990 as Problem~535 in \emph{Open Problems in Topology}: for a core-compact space $X$ and a dcpo $P$ equipped with its Scott topology, under what conditions on $P$ do the Isbell and Scott topologies on $C(X,P)$ agree?It was proved that, for a nonempty bicomplete continuous dcpo $P$, the Isbell and Scott topologies on $C(X,P)$ coincide for every core-compact space $X$ if and only if $P$ is bounded complete; for every compact core-compact space $X$ if and only if $P$ is conditionally bounded complete; and for every RW-space $X$ if and only if $P$ is a pointed continuous $L$-domain.We remove the bicompleteness assumption from all three classifications by combining the forbidden-retract theorem of Jia, Jung and Li with a separation theorem for powers of downward well-ordered chains and suitable Alexandrov test spaces.

math.GN↗

arXiv2026

Three Problems on Separable Quotients of Precompact Abelian Groups

We address three problems on separable quotients of topological groups posed by Leiderman, Morris, and Tkachenko in \cite{LMT} published on Israel Journal of Mathematics. First, we construct in ZFC a connected Baire Pontryagin-reflexive dense subgroup of $\T^{\cc}$ whose countable subgroups are $h$-embedded and whose uncountable subgroups are dense. Its underlying abstract group is the circle group, and all its compact subsets are finite. Second, we construct a zero-dimensional Baire Pontryagin-reflexive example with the same subgroup properties whose underlying group is free abelian of rank $\cc$. Both examples have no nontrivial separable Hausdorff quotient. Third, for the group constructed in their Theorem~3.5, we determine every closed subgroup of every finite power up to an integral change of coordinates and prove that every countable subgroup of every Hausdorff quotient of a finite power is $h$-embedded and closed. The same conclusions hold for our free Baire reflexive example. These results answer Problem~1.25 negatively, Problem~3.12 affirmatively and realize all three regularity properties in Problem~3.14 simultaneously in \cite{LMT}.

math.GN↗