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• W2089391331 abstract "The authors construct a completely regular space Xn such that Player I has a winning strategy in the point-open game G(Xn), but Xo has no <7-closure-preserving cover by C-scattered closed sets. The purpose of this paper is to present the following. EXAMPLE. A completely regular space Xo such that Player I has a winning strategy in the point-open game G(Xn), but X0 has no cr-closure-preserving cover by C-scattered closed sets. Given a space X, the point-open game G(X) is defined as follows: at move n, Player I chooses a point xn in X, and then Player II chooses an open neighborhood Un of xn. Player I wins the play (xo, Uo, xi, U,... ) of G(X) if (J{Un : n < oj} — X (see (G, T2)). If instead of points xn Player I chooses compact sets C, and if Player I has a winning strategy in the resulting game, then X is called compact- like (see (T2, T3)). If a point x in a space X has an open neighborhood U such that U is compact, then x is called a point of local compactness of X. A space X such that each nonempty closed subspace has a point of local compactness is called C-scattered (Tj.). A family J of subsets of X with the property that for any subfamily of 7, the closure and the sum commute, is called closure-preserving (M). A countable union of closure-preserving families is called cr-closure-preserving. The space Xo in the above example is a modification of the space constructed by Nogura (N); however, it improves his result in several aspects. Nogura showed, un- der the continuum hypotheses, that a compact-like space need not have a countable cover by C-scattered closed subsets, answering a question of Telgarsky (T2, T3). Here, (1) the continuum hypothesis is eliminated, (2) the game condition involving Player I is substantially strengthened, and (3) the countable union is generalized to a rr-closure-preserving union. Since each compact space is C-scattered, it follows that Xo has no cr-closure- preserving cover by compact sets. In contrast, a hereditarily metacompact compact- like space does have a closure-preserving cover by compact sets (see (JST)). More- over, each compact-like space is the union of countably many C-scattered subsets (see (T3)), where the C-scattered subsets cannot be closed in general (it follows from (N) and also from the above example)." @default.
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• W2089391331 date "1987-03-01" @default.
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• W2089391331 title "A singular space related to the point-open game" @default.
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• W2089391331 doi "https://doi.org/10.1090/s0002-9939-1987-0875399-6" @default.
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