On certain localized version of uniform selection principles

TL;DR

This paper introduces local variations of uniform selection principles in uniform spaces, explores their properties, relationships, and provides examples to illustrate their distinct behaviors.

Contribution

It defines and studies locally $mbda$-bounded spaces as local versions of uniform selection principles, analyzing their properties and interrelations.

Findings

Established the concept of locally $mbda$-bounded spaces.

Analyzed preservation properties and critical cardinals related to these local principles.

Provided examples demonstrating the distinct behaviors of the new notions.

Abstract

We intend to localize the selection principles in uniform spaces (Ko\v{c}inac, 2003) by introducing their local variations, namely locally $\Upsilon$-bounded spaces (where $\Upsilon$ 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.