On $\theta$-Hurewicz and $\alpha$-Hurewicz Topological spaces

TL;DR

This paper introduces and explores new variants of Hurewicz properties in topological spaces, examining their relationships, equivalences in specific spaces, and behavior under mappings, with implications for boundedness in certain subspaces.

Contribution

It defines $ heta$-Hurewicz and $eta$-Hurewicz properties, investigates their equivalences in extremally disconnected spaces, and studies their preservation under mappings and implications for boundedness.

Findings

In extremally disconnected semi-regular spaces, multiple Hurewicz properties are equivalent.

Finite powers of extremally disconnected spaces have $ heta$-Hurewicz property iff they satisfy a specific selection principle.

Mildly Hurewicz subspaces of $oldsymbol{ ext{ extomega}}^ ext{ extomega}$ are bounded.

Abstract

In this paper, we introduced $\alpha$-Hurewicz $\&$ $\theta$-Hurewicz properties in a topological space $X$ and investigated their relationship with other selective covering properties. We have shown that for an extremally disconnected semi-regular spaces, the properties: Hurewicz, semi-Hurewicz, $\alpha$-Hurewicz, $\theta$-Hurewicz, almost-Hurewicz, nearly Hurewicz and midly Hurewicz are equivalent. We have also proved that for an extremally disconnected space X, every finite power of X has $\theta$-Hurewicz property if and only if X has the selection principle $U_{fin}(\theta$-$\Omega, \theta$-$\Omega)$. The preservation under several types of mappings of $\alpha$-Hurewicz and $\theta$-Hurewicz properties are also discussed. Also, we showed that, if $X$ is a mildly Hurewicz subspace of $ \omega^\omega$, than $X$ is bounded.