Skeletal maps and \(I\)-favorable spaces. (English) Zbl 1228.54018

A continuous surjection \(f:X\to Y\) is said to be skeletal, provided the closure of \(f(U)\) has a nonempty interior for each open \(U\subseteq X\). A topological space \((X,\mathcal T)\) is called I-favorable (i.e. player I has a winning strategy in the open-open game introduced by P. Daniels, K. Kunen and H. Zhou in [Fundam. Math. 145, No. 3, 205–220 (1994; Zbl 0811.54008)]), provided there is a function \(\sigma:\mathcal T^{<\omega}\to\mathcal T\) such that whenever a sequence \((B_n)_n\) of nonempty elements of \(\mathcal T\) satisfies \(\sigma(\emptyset)=B_0\), and \(B_{n+1}\subseteq \sigma(B_0,\dots,B_n)\) for each \(n\in\omega\), then \(\bigcup_{n\in\omega} B_n\) is dense in \(X\).
The authors show that the compact I-favorable spaces and skeletal maps form an adequate pair in the sense of E. V. Shchepin [Usp. Mat. Nauk 31, No. 5 (191), 191–226 (1976; Zbl 0345.54022)]. In particular, the class of compact I-favorable spaces is the smallest class which contains all metric compact spaces and is closed under skeletal maps, as well as under limits of \(\sigma\)-complete inverse systems with skeletal bonding maps.


54C10 Special maps on topological spaces (open, closed, perfect, etc.)
91A44 Games involving topology, set theory, or logic
54C35 Function spaces in general topology
54B35 Spectra in general topology
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