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On (strong) $α$-favorability of the Vietoris hyperspace

In: Mathematica Slovaca, vol. 63, no. 2
Leszek Piątkiewicz - László Zsilinszky
Detaily:
Rok, strany: 2013, 321 - 330
Kľúčové slová:
Vietoris topology, Banach-Mazur game, strong Choquet game, (strongly) (weakly) $\alpha$-favorable space, Baire space, Tychonoff plank, Bernstein set
O článku:
For a normal space $X$, $α$ (i.e. the nonempty player) having a winning strategy (resp. winning tactic) in the strong Choquet game $Ch(X)$ played on $X$ is equivalent to $α$ having a winning strategy (resp. winning tactic) in the strong Choquet game played on the hyperspace $CL(X)$ of nonempty closed subsets endowed with the Vietoris topology $τV$. It is shown that for a non-normal $X$ where $α$ has a winning strategy (resp. winning tactic) in $Ch(X)$, $α$ may or may not have a winning strategy (resp. winning tactic) in the strong Choquet game played on the Vietoris hyperspace. If $X$ is quasi-regular, then having a winning strategy (resp. winning tactic) for $α$ in the Banach-Mazur game $BM(X)$ played on $X$ is sufficient for $α$ having a winning strategy (resp. winning tactic) in $BM(CL(X),τV)$, but not necessary, not even for a separable metric $X$. In the absence of quasi-regularity of a space $X$ where $α$ has a winning strategy in $BM(X)$, $α$ may or may not have a winning strategy in the Banach-Mazur game played on the Vietoris hyperspace.
Ako citovať:
ISO 690:
Piątkiewicz, L., Zsilinszky, L. 2013. On (strong) $α$-favorability of the Vietoris hyperspace. In Mathematica Slovaca, vol. 63, no.2, pp. 321-330. 0139-9918. DOI: https://doi.org/10.2478/s12175-012-0100-3

APA:
Piątkiewicz, L., Zsilinszky, L. (2013). On (strong) $α$-favorability of the Vietoris hyperspace. Mathematica Slovaca, 63(2), 321-330. 0139-9918. DOI: https://doi.org/10.2478/s12175-012-0100-3
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