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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.

MSC:

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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