Ideal analogues of some variants of Hurewicz property

Brij K. Tyagi, Sumit singh, Manoj Bhardwaj


In this paper, we continue the study on the ideal analogues of several  variations of Hurewicz property introduced by Das et al. \cite{Das,DKC,DCS} for example, $  \mathcal{I} $-Hurewicz ($  \mathcal{I} $H), star $  \mathcal{I} $-Hurewicz (S$  \mathcal{I} $H), weakly $ \mathcal{I} $-Hurewicz (W$  \mathcal{I} $H) and weakly star-$ \mathcal{I} $-Hurewicz (WS$  \mathcal{I} $H). It is shown that several implications in the relationship diagram of their concepts are reversible under certain conditions, for instance; (1) If paracompact Hausdorff space has the WS$ \mathcal{I} $H property then  it has W$ \mathcal{I} $H property. (2)If the complement of dense set has $ \mathcal{I} $H property then W$ \mathcal{I} $H property implies $ \mathcal{I} $H property and(3)If the complement of dense set has S$ \mathcal{I} $H property then WS$ \mathcal{I} $H property implies S$ \mathcal{I} $H property. Examples of spaces are given which have W$ \mathcal{I} $H property but not separable.  In addition we introduced the ideal analogues of some new variations of Hurewicz property called mildly $ \mathcal{I} $-Hurewicz and star K-$ \mathcal{I} $-Hurewicz properties  and explore their relationship with other variations of $ \mathcal{I} $-Hurewicz property. We also study the preservation under certain mappings.