Countably compact spaces admitting full \(r\)-skeletons are proximal
A partially ordered set \((\Gamma, \leq)\) is up-directed if, for any \(s,t\in \Gamma\), there exists \(u\in \Gamma\) such that \(s\leq u\) and \(t\leq u\). The set \(\Gamma\) is countably complete if, for any sequence \(\{t_n: n\in\omega\}\subset \Gamma\) such that \(t_n\leq t_{n+1}\) for any \(n\in\omega\), there exists \(\sup\{x_n: n\in\omega\}\). A full \(r\)-skeleton in a space \(X\) is a family \(\{r_s:s\in \Gamma\}\) of retractions on \(X\) with the following properties: \medskip \noindent (r1) \(\Gamma\) is a partially ordered, up-directed and countably complete set; \smallskip \noindent (r2) \(r_s(X)\) has a countable network for each \(s\in \Gamma\) and \(\bigcup_{s\in \Gamma}r_s(X)=X\); \smallskip \noindent (r3) if \(s,t\in \Gamma\) and \(s\leq t\), then \(r_s\circ r_t= r_t\circ r_s=r_s\); \smallskip \noindent (r4) if \(\{s_n\}_{n\in\omega}\subset \Gamma\), \(s_n\leq s_{n+1}\) for each \(n\in\omega\) and \(t=\sup_{n\in\omega}s_n\), then \(r_t(x) =\lim r_{s_n}(x)\) for any \(x\in X\). A proximal game on a space \(X\) is defined for Players \(N\) and \(P\) and has \(\omega\)-many rounds. In the \(n\)-th round, Player \(N\) chooses a symmetric open neighborhood \(U_n\) of the diagonal \(\Delta=\{(x,x): x\in X\}\) of the space \(X\) and \(P\) picks a point \(x_n\in X\) in such a way that \(x_{k+1}\in U_k[x_k]\) for any \(k\in\omega\). Player \(N\) wins if either \(\bigcap\{U_n[x_n]: n\in\omega\}=\emptyset\) or the sequence \((x_n)\) converges. Otherwise \(P\) is declared to be the winner of the play \(\{U_n, x_n: n\in\omega\}\). The space \(X\) is called proximal if Player \(N\) has a winning strategy in the proximal game on \(X\). The main result of the paper states that any countably compact Tychonoff space, that admits a full \(r\)-skeleton, is proximal. The authors also introduce and study the notions of full \(c\)-skeletons and full \(q\)-skeletons. They prove that the respective classes of spaces have nice categorical properties and a countably compact space has a full \(c\)-skeleton if and only if it has a full \(r\)-skeleton.
scientific article; zbMATH DE number 7375604
scientific article