Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2112.13106

On certain localized version of uniform selection principles

Nur Alam·Debraj Chandra
View original source ↗Read source document ↗Export metadata ↓

Abstract

We intend to localize the selection principles in uniform spaces (Kočinac, 2003) by introducing their local variations, namely locally $Υ$-bounded spaces (where $Υ$ is Menger, Hurewicz or Rothberger). It has been observed that the difference between uniform selection principles and the corresponding local correlatives as introduced here is reasonable enough to discuss about these new notions. Certain observations using the critical cardinals (on the uniform selection principles which have not studied before) as well as preservation like properties (on the local versions) are presented. The interrelationships between the notions considered in this paper are outlined into an implication diagram. Certain interactions between these local variations are also investigated. We present several examples to illustrate the distinguishable behaviour of the new notions.

Explore related subjects

math.GN

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Nur Alam, Debraj Chandra. 2022-12-28. On certain localized version of uniform selection principles. https://doi.org/10.2298/fil2220855a

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

KEEP EXPLORING

Related papers

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↗
arXiv2026

Topological Vector Group Topologies Between the Minimal Topology and the Usual Topology on the Real Line

For every positive sequence that tends to zero faster than every fixed exponential, we construct a Hausdorff topological Vector Group topology on the additive group of real numbers. It lies strictly between the minimal Hausdorff topological Vector Group topology and the usual topology. The construction is illustrated by factorial powers, quadratic exponential decay, and prime radicals divided by a quadratic exponential. We also record a finite scalar covering criterion for comparing two such topologies.

math.GN↗
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↗