On traditional Menger and Rothberger variations (Q6640036)

A comprehensive survey of various Menger and Rothberger selection properties involving as few separation properties as possible is presented. Care is also taken to specify the relevant open covers: in particular it is noted that different authors do not agree on what constitutes \(\Omega\), `the' family of \(\omega\)-covers. For two selection games \(\mathcal {G,H}\), the notation \(\mathcal G\le_{\mathrm{II}}^+\mathcal H\) is defined to mean that if player II has a (Markov) winning strategy for \(\mathcal G\) then II also has a (Markov) winning strategy for \(\mathcal H\), but if player I does not have a (predetermined, constant) winning strategy for \(\mathcal G\) then player I does not have a (predetermined, constant) winning strategy for \(\mathcal H\). With no separation assumptions it is shown that \(\mathsf G_{\mathrm{fin}}(\mathcal{K,K})\le_{\mathrm{II}}^+\mathsf G_{\mathrm{fin}}(\Omega,\Omega)\le_{\mathrm{II}}^+\mathsf G_{\mathrm{fin}}(\mathcal{O,O})\) but only the second of these holds for the game \(\mathsf G_1\), where \(\mathsf G\) is the standard selection game. Numerous other results, including finite productivity, as well as examples illustrating irreversibility of certain implications are presented.