Mildly version of Hurewicz basis covering property and Hurewicz measure zero spaces (Q6073785)

This paper studies variations on the Hurewicz basis covering property. Specifically, the authors study what they term the ``mildly Hurewicz property'' (MH): A space \(X\) is said to have the \textit{mild 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 \(n\in\omega\), \(\mathcal V_n\) is a finite subset of \(\mathcal U_n\) and each \(x\in X\) belongs to \(\bigcup \mathcal V_n\) for all but finitely many \(n\in\omega\). When \(\mathcal A\) and \(\mathcal B\) denote families of subsets of a set \(S\), the notation \(CDR_{sub}(\mathcal A,\mathcal B)\) denotes the statement given any sequence \(\langle A_n \rangle \) of elements of \(\mathcal A\) there is a sequence \(\langle B_n \rangle \) of elements of \(\mathcal B\) such that for all \(n\in\omega\), \(B_n\subseteq A_n\) and for distinct \(m,n\in\omega\), \(B_m\cap B_n=\emptyset\). In this work, an important case is when both \(\mathcal A\) and \(\mathcal B\) are the collection \(\mathcal C_\mathcal O\) of all clopen covers of a metric space \(X\). After an introductory section and a section of new definitions, the remaining two sections of this paper give a number of conditions equivalent to the mild Hurewicz property. To mention just one such theorem, it is shown that if \(CDR_{sub}(\mathcal C_\mathcal O,\mathcal C_\mathcal O)\) holds then having the mild Hurewicz property is equivalent to having the mild Hurewicz basis property. Many other equivalences of this type are given, but all involve new and rather complicated definitions.