View source for Sierpiński game

Let $Y$ be a [[topological space]] and $X$ an uncountable subset of $Y$. Two players alternatively select subsets of $X$. Player I selects some uncountable subset $A_1$ of $X$. Player II answers by picking up an uncountable subset $B_1 \subset A_1$. Then again player I selects some uncountable set $A_2 \subset B_1$ and player II responds by selecting some uncountable subset $B_2 \subset A_2$. Playing this way the two players generate a decreasing sequence $P = (A_i,B_i)_{i \ge 1}$ of uncountable sets, which is called a play. By definition, player II wins this play if the intersection $\cap \bar B_i$ of the closures of $B_i$ in $Y$ is contained in $X$. 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 $Y$ 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 $X$ contains the [[Cantor discontinuum]] (and therefore contains continuum many points). On the other hand, if $X$ is a Suslin subset of $Y$ (cf. also [[Descriptive set theory]]), then player II has a winning strategy ([[#References|[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 ([[#References|[a1]]] and [[Borel set]]) and F. Hausdorff ([[#References|[a2]]]) when they were verifying the truth of the [[continuum hypothesis]] for such subsets of the unit segment. W. Sierpiński ([[#References|[a5]]]) gave another proof of the same result. It was this proof of Sierpiński that made R. Telgársky ([[#References|[a6]]]) introduce the above game and name it after Sierpiński. Further information concerning the game of Sierpiński can be found in [[#References|[a3]]], [[#References|[a4]]] and [[#References|[a7]]].

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  P.S. Alexandrov,   "Sur la puissance des ensembles mesurables B"  ''C.R. Acad. Sci. Paris'' , '''162'''  (1916)  pp. 323–325</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  F. Hausdorff,   "Die Mächtigkeit der Borelschen Mengen"  ''Math. Ann.'' , '''77'''  (1916)  pp. 430–437</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  G. Kubicki,   "On a game of Sierpiński"  ''Colloq. Math.'' , '''54'''  (1987)  pp. 179–192</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  G. Kubicki,   "On a modified game of Sierpiński"  ''Colloq. Math.'' , '''53'''  (1987)  pp. 81–91</TD></TR>
<TR><TD valign="top">[a5]</TD> <TD valign="top">  W. Sierpiński,   "Sur le puissance des ensembles mesurables (B)"  ''Fundam. Math.'' , '''5'''  (1924)  pp. 166–171</TD></TR>
<TR><TD valign="top">[a6]</TD> <TD valign="top">  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</TD></TR>
<TR><TD valign="top">[a7]</TD> <TD valign="top">  R. Telgárski,   "Topological games: On the 50th anniversary of the Banach–Mazur game"  ''Rocky Mount. J. Math.'' , '''17'''  (1987)  pp. 227–276</TD></TR>
</table>

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