Sierpiński game
Let be a topological space and
an uncountable subset of
. Two players alternatively select subsets of
. Player I selects some uncountable subset
of
. Player II answers by picking up an uncountable subset
. Then again player I selects some uncountable set
and player II responds by selecting some uncountable subset
. Playing this way the two players generate a decreasing sequence
of uncountable sets, which is called a play. By definition, player II wins this play if the intersection
of the closures of
(in
) is contained in
. Otherwise the play is won by player I. A given "rule" of selecting the moves of player II is called a winning strategy for player II if every play generated by this rule is won by this player.
If is a Polish space (a completely metrizable and separable space, cf. also Vague topology; Descriptive set theory; Complete metric space; Separable space), then the existence of a winning strategy for player II implies that
contains the Cantor discontinuum (and therefore contains continuum many points). On the other hand, if
is a Suslin subset of
(cf. also Descriptive set theory), then player II has a winning strategy ([a3]). Thus, every uncountable Suslin subset of a Polish space contains the Cantor discontinuum. For Borel subsets of the unit segment this was proved by P.S. Aleksandrov ([a1] and Borel set) and F. Hausdorff ([a2]) when they were verifying the truth of the continuum hypothesis for such subsets of the unit segment. W. Sierpiński ([a5]) gave another proof of the same result. It was this proof of Sierpiński that made R. Telgársky ([a6]) introduce the above game and name it after Sierpiński. Further information concerning the game of Sierpiński can be found in [a3], [a4] and [a7].
References
[a1] | P.S. Alexandrov, "Sur la puissance des ensembles mesurables B" C.R. Acad. Sci. Paris , 162 (1916) pp. 323–325 |
[a2] | F. Hausdorff, "Die Mächtigkeit der Borelschen Mengen" Math. Ann. , 77 (1916) pp. 430–437 |
[a3] | G. Kubicki, "On a game of Sierpiński" Colloq. Math. , 54 (1987) pp. 179–192 |
[a4] | G. Kubicki, "On a modified game of Sierpiński" Colloq. Math. , 53 (1987) pp. 81–91 |
[a5] | W. Sierpiński, "Sur le puissance des ensembles mesurables (B)" Fundam. Math. , 5 (1924) pp. 166–171 |
[a6] | R. Telgárski, "On some topological games" , Proc. Fourth Prague Topological Symp. 1976, Part B: Contributed papers , Soc. Czech. Math. and Physicists (1977) pp. 461–472 |
[a7] | R. Telgárski, "Topological games: On the 50th anniversary of the Banach–Mazur game" Rocky Mount. J. Math. , 17 (1987) pp. 227–276 |
Sierpiński game. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sierpi%C5%84ski_game&oldid=12352