The Banach-Mazur game and generic existence of solutions to optimization problems

Kenderov, P. S.; Revalski, J. P.

doi:10.1090/S0002-9939-1993-1137224-0

The Banach-Mazur game and generic existence of solutions to optimization problems
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by P. S. Kenderov and J. P. Revalski PDF

Proc. Amer. Math. Soc. 118 (1993), 911-917 Request permission

Abstract:

The existence of a winning strategy in the well-known Banach-Mazur game in a completely regular topological space $X$ is proved to be equivalent to the generic existence of solutions of optimization problems generated by continuous functions in $X$.

References

Lectures on analysis

M. M. Čoban and P. S. Kenderov, Dense Gâteaux differentiability of the sup-norm in $C(T)$ and the topological properties of $T$, C. R. Acad. Bulgare Sci. 38 (1985), no. 12, 1603–1604. MR 837262

Generic Gâteaux differentiability of convex functionals in

and the topological properties of

M. M. Čoban, P. S. Kenderov, and J. P. Revalski, Generic well-posedness of optimization problems in topological spaces, Mathematika 36 (1989), no. 2, 301–324 (1990). MR 1045790, DOI 10.1112/S0025579300013152
Gabriel Debs, Stratégies gagnantes dans certains jeux topologiques, Fund. Math. 126 (1985), no. 1, 93–105 (French, with English summary). MR 817083, DOI 10.4064/fm-126-1-93-105
J. E. Jayne and C. A. Rogers, Borel selectors for upper semicontinuous set-valued maps, Acta Math. 155 (1985), no. 1-2, 41–79. MR 793237, DOI 10.1007/BF02392537
I. Namioka, Radon-Nikodým compact spaces and fragmentability, Mathematika 34 (1987), no. 2, 258–281. MR 933504, DOI 10.1112/S0025579300013504
John C. Oxtoby, The Banach-Mazur game and Banach category theorem, Contributions to the theory of games, vol. 3, Annals of Mathematics Studies, no. 39, Princeton University Press, Princeton, N.J., 1957, pp. 159–163. MR 0093741
N. K. Ribarska, Internal characterization of fragmentable spaces, Mathematika 34 (1987), no. 2, 243–257. MR 933503, DOI 10.1112/S0025579300013498
Charles Stegall, Topological spaces with dense subspaces that are homeomorphic to complete metric spaces and the classification of $C(K)$ Banach spaces, Mathematika 34 (1987), no. 1, 101–107. MR 908845, DOI 10.1112/S0025579300013334
Rastislav Telgársky, Topological games: on the 50th anniversary of the Banach-Mazur game, Rocky Mountain J. Math. 17 (1987), no. 2, 227–276. MR 892457, DOI 10.1216/RMJ-1987-17-2-227
A. N. Tihonov, Stability of a problem of optimization of functionals, Ž. Vyčisl. Mat i Mat. Fiz. 6 (1966), 631–634 (Russian). MR 198308
H. E. White Jr., Topological spaces that are $\alpha$-favorable for a player with perfect information, Proc. Amer. Math. Soc. 50 (1975), 477–482. MR 367941, DOI 10.1090/S0002-9939-1975-0367941-0