On localization of the Menger property

TL;DR

This paper introduces and explores the locally Menger property in topology, examining its preservation, differences from the classical Menger property, and related concepts like MG-spaces, with various topological insights.

Contribution

It defines the locally Menger property, investigates its behavior, introduces MG-spaces, and studies their topological characteristics and differences from classical properties.

Findings

Locally Menger spaces have unique preservation properties.

Differences between local and classical Menger properties are identified.

Topological features like character, network weight, and decomposition are analyzed.

Abstract

In this paper we introduce and study the local version of the Menger property, namely locally Menger property (or, locally Menger space). We explore some preservation like properties in this space. We also discuss certain situations where this local property behaves somewhat differently from the classical Menger property. Some observations about the character of a point, network weight and weight in this space are also investigated carefully. We also introduce the notion of Menger generated space (in short, MG-space) and make certain investigations in this space. Several topological observations on the decomposition and the remainder of locally Menger spaces are also discussed.