Mildly version of Hurewicz Basis covering property and Hurewicz measure zero spaces

TL;DR

This paper introduces a new mildly-Hurewicz property for topological spaces, characterizes it via basis and measure zero properties, and extends the classical Hurewicz covering property to a milder form.

Contribution

It defines the mildly-Hurewicz property, relates it to basis and measure zero properties, and provides characterizations for metrizable spaces.

Findings

Mildly-Hurewicz property is characterized by specific basis and measure zero conditions.

The property generalizes the classical Hurewicz covering property.

Characterizations are established for metrizable spaces.

Abstract

In this paper, we introduced the mildly version of the Hurewicz basis covering property, studied by Babinkostova, Ko\v{c}inac, and Scheepers. A space $X$ is said to have mildly-Hurewicz property if for each sequence $\langle \mathcal{U}_n : n\in \omega \rangle$ of clopen covers of $X$ there is a sequence $\langle \mathcal{V}_n : n\in \omega \rangle$ such that for each $n$, $\mathcal{V}_n$ is a finite subset of $\mathcal{U}_n$ and for each $x\in X$, $x$ belongs to $\bigcup\mathcal{V}_n$ for all but finitely many $n$. Then we characterized mildly-Hurewicz property by mildly-Hurewicz Basis property and mildly-Hurewicz measure zero property for metrizable spaces.