On $\hat{I}¸$-Hurewicz and $\hat{I}\pm$-Hurewicz Topological spaces

Abstract

In this paper, we introduced $\hat{I}\pm$-Hurewicz $\&$ $\hat{I}¸$-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, $\hat{I}\pm$-Hurewicz, $\hat{I}¸$-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 $\hat{I}¸$-Hurewicz property if and only if X has the selection principle $U_{fin}(\hat{I}¸$-$\hat{I}\text{c◯},\hat{I}¸$-$\hat{I}\text{c◯})$. The preservation under several types of mappings of $\hat{I}\pm$-Hurewicz and $\hat{I}¸$-Hurewicz properties are also discussed. Also, we showed that, if $X$ is a mildly Hurewicz subspace of $\ddot{I}{‰}^{\ddot{I}}‰$, than $X$ is bounded.