Game theoretic approach to skeletally Dugundji and Dugundji spaces (Q5963967)

The authors provide characterizations of \textit{(skeletally) Dugundji spaces} in terms of club collections consisting of countable families of co-zero sets. To review the main results, recall that a family \(\Psi\) of continuous maps with a common domain \(X\) is a \textit{multiplicative lattice} (\textit{\(\omega\)-multiplicative lattice}, resp.), provided (1) whenever \(f:X\to f(X)\) (resp., \(f(X)\) is 2nd countable) there is \(\phi\in\Psi\) so that \(f=h\circ\phi\) for some continuous \(h:\phi(X)\to f(X)\), and the weight of \(\phi(X)\) is not more than that of \(f(X)\); (2) the diagonal map of a (countable) subcollection of \(\Psi\) is homeomorphic to some element of \(\Psi\). A collection \(\mathcal C\) of countably infinite families of co-zero sets of a Tychonoff space \(X\) is said to be a \textit{\(c\)-club} (resp., \textit{\(d\)-club}) provided (i) \(\mathcal C\) is closed under increasing \(\omega\)-chains, (ii) each countable collection of co-zero sets is contained in an element of \(\mathcal C\), (iii) each \(\mathcal P\in\mathcal C\) is closed under finite unions and finite intersections, for each \(W\in \mathcal P\) there are subsets of \(\mathcal P\), \(\{U_n\}, \{V_n\}\) such that \(U_n\subseteq X\setminus V_n\subseteq U_{n+1}\) for all \(n\), and \(W=\bigcup\{U_n:n\geq 0\}\), moreover, for each nonempty co-zero set \(V\) there exists \(W\in \mathcal P\) such that \(U\in\mathcal P\) and \(U\subseteq W\) imply \(U\cap V\neq\emptyset\) (resp., for any \(\mathcal S\subset\mathcal P\) and \(x\notin \overline{\bigcup\mathcal S}\), there is \(W\in\mathcal P\) with \(x\in W\) and \(W\cap \bigcup\mathcal S=\emptyset\)). A map \(f:X\to Y\) is \textit{skeletal} (resp., \textit{\(d\)-open}) provided \(\text{int}\overline{f(U)}\neq \emptyset\) (resp., \(f(U)\subseteq \text{int}\overline{f(U)}\)) for every nonempty open \(U\subseteq X\).NEWLINENEWLINEThe main result of the paper states that for a Tychonoff space \(X\) the following are equivalent: (1) \(X\) is skeletally Dugundji, (2) \(X\) has a (\(\omega\)-) multiplicative lattice of skeletal maps, (3) \(X\) has an additive \(c\)-club. Also, the existence of an additive \(d\)-club is equivalent to having a (\(\omega\)-) multiplicative lattice of \(d\)-open maps, which results in a characterization of compact Hausdorff Dugundji spaces as the ones possessing an additive \(d\)-club.