ON θ-HUREWICZ AND α-HUREWICZ TOPOLOGICAL SPACES
Abstract
In this paper, we introduced $\alpha$-Hurewicz and %$\&$
$\theta$-Hurewicz properties in a topological space $X$ and investigated their relationship with other selective covering properties. We have shown that for any extremally disconnected semi-regular spaces,
the properties: Hurewicz, semi-Hurewicz$,\alpha$-Hurewicz, $\theta$-Hurewicz, almost-Hurewicz, nearly Hurewicz and mildly Hurewicz are equivalent. We have also proved that for an extremally disconnected space X, every finite power of X has the $\theta$-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 have shown that if $X$ is a mildly Hurewicz subspace
of $\omega^\omega$, than $X$ is bounded.
$\theta$-Hurewicz properties in a topological space $X$ and investigated their relationship with other selective covering properties. We have shown that for any extremally disconnected semi-regular spaces,
the properties: Hurewicz, semi-Hurewicz$,\alpha$-Hurewicz, $\theta$-Hurewicz, almost-Hurewicz, nearly Hurewicz and mildly Hurewicz are equivalent. We have also proved that for an extremally disconnected space X, every finite power of X has the $\theta$-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 have shown that if $X$ is a mildly Hurewicz subspace
of $\omega^\omega$, than $X$ is bounded.