Banach-Mazur game
A game that appeared in the famous Scottish Book [a11], [a6], where its initial version was formulated as Problem 43 by the Polish mathematician S. Mazur: Given the space of real numbers and a non-empty subset
of it, two players
and
play a game in the following way:
starts by choosing a non-empty interval
of
and then
responds by choosing a non-empty subinterval
of
. Then player
in turn selects a non-empty interval
and
continues by taking a non-empty subinterval
of
. This procedure is iterated infinitely many times. The resulting infinite sequence of nested intervals
is called a play. By definition, the player
wins this play if the intersection
has a common point with
. Otherwise
wins. Mazur had observed two facts:
a) if the complement of in some interval of
is of the first Baire category in this interval (equivalently, if
is residual in some interval of
, cf. also Category of a set; Baire classes), then player
has a winning strategy (see below for the definition); and
b) if itself is of the first Baire category in
, then
has a winning strategy. The question originally posed by Mazur in Problem 43 of the Scottish Book (with as prize a bottle of wine!) was whether the inverse implications in the above two assertions hold. On August 4, 1935, S. Banach wrote in the same book that "Mazur's conjecture is true" . The proof of this statement of Banach however has never been published. The game subsequently became known as the Banach–Mazur game.
More than 20 years later, in 1957, J. Oxtoby [a8] published a proof for the validity of Mazur's conjecture. Oxtoby considered a much more general setting. The game was played in a general topological space with
and the two players
and
were choosing alternatively sets
from an a priori prescribed family of sets
which has the property that every element of
contains a non-empty open subset of
and every non-empty open subset of
contains an element of
. As above,
wins if
, otherwise
wins. Oxtoby's theorem says that
has a winning strategy if and only if
is of the first Baire category in
; also, if
is a complete metric space, then
has a winning strategy exactly when
is residual in some non-empty open subset of
.
Later, the game was subjected to different generalizations and modifications.
Generalizations.
Only the most popular modification of this game will be considered. It has turned out to be useful not only in set-theoretic topology, but also in the geometry of Banach spaces, non-linear analysis, number theory, descriptive set theory, well-posedness in optimization, etc. This modification is the following: Given a topological space , two players (usually called
and
) alternatively choose non-empty open sets
(in this sequence the
are the choices of
and the
are the choices of
; thus, it is player
who starts this game). Player
wins the play
if
, otherwise
wins. To be completely consistent with the general scheme described above, one may think that
and
starts by always choosing the whole space
. This game is often denoted by
.
A strategy for the player
is a mapping which assigns to every finite sequence
of legal moves in
a non-empty open subset
of
included in the last move of
(i.e.
). A stationary strategy (called also a tactics) for
is a strategy for this player which depends only on the last move of the opponent. A winning strategy (a stationary winning strategy)
for
is a strategy such that
wins every play in which his/her moves are obtained by
. Similarly one defines the (winning) strategies for
.
A topological space is called weakly
-favourable if
has a winning strategy in
, while it is termed
-favourable if there is a stationary winning strategy for
in
. It can be derived from the work of Oxtoby [a8] (see also [a4], [a7] and [a9]) that the space
is a Baire space exactly when player
does not have a winning strategy in
. Hence, every weakly
-favourable space is a Baire space. In the class of metric spaces
, a metric space is weakly
-favourable if and only if it contains a dense and completely metrizable subspace. One can use these two results to see that the Banach–Mazur game is "not determined" . I.e. it could happen for some space
that neither
nor
has a winning strategy. For instance, the Bernstein set
in the real line (cf. also Non-measurable set) is a Baire space which does not contain a dense completely metrizable subspace (consequently
does not admit a winning strategy for either
or
).
The above characterization of weak -favourability for metric spaces has been extended for some non-metrizable spaces in [a10].
A characterization of -favourability of a given completely-regular space
can be obtained by means of the space
of all continuous and bounded real-valued functions on
equipped with the usual sup-norm
. The following statement holds [a5]: The space
is weakly
-favourable if and only if the set
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is residual in . In other words,
is weakly
-favourable if and only if almost-all (from the Baire category point of view) of the functions in
attain their maximum in
. The rich interplay between
,
and
is excellently presented in [a3].
The class of -favourable spaces (spaces which admit
-winning tactics) is strictly narrower than the class of weakly
-favourable spaces. G. Debs [a2] has exhibited a completely-regular topological space
which admits a winning strategy for
in
, but does not admit any
-winning tactics in
. Under the name "espaces tamisables" , the
-favourable spaces were introduced and studied also by G. Choquet [a1].
[a10] is an excellent survey paper about topological games (including ).
References
[a1] | G. Choquet, "Une classe régulières d'espaces de Baire" C.R. Acad. Sci. Paris Sér. I , 246 (1958) pp. 218–220 |
[a2] | G. Debs, "Strategies gagnantes dans certain jeux topologique" Fundam. Math. , 126 (1985) pp. 93–105 |
[a3] | G. Debs, J. Saint-Raymond, "Topological games and optimization problems" Mathematika , 41 (1994) pp. 117–132 |
[a4] | M.R. Krom, "Infinite games and special Baire space extensions" Pacific J. Math. , 55 : 2 (1974) pp. 483–487 |
[a5] | P.S. Kenderov, J.P. Revalski, "The Banach–Mazur game and generic existence of solutions to optimization problems" Proc. Amer. Math. Soc. , 118 (1993) pp. 911–917 |
[a6] | "The Scottish Book: Mathematics from the Scottish Café" R.D. Mauldin (ed.) , Birkhäuser (1981) |
[a7] | R.A. McCoy, "A Baire space extension" Proc. Amer. Math. Soc. , 33 (1972) pp. 199–202 |
[a8] | J. Oxtoby, "The Banach–Mazur game and the Banach category theorem" , Contributions to the Theory of Games III , Ann. of Math. Stud. , 39 , Princeton Univ. Press (1957) pp. 159–163 |
[a9] | J. Saint-Raymond, "Jeux topologiques et espaces de Namioka" Proc. Amer. Math. Soc. , 87 (1983) pp. 499–504 Zbl 0511.54007 |
[a10] | R. Telgárski, "Topological games: On the fifth anniversary of the Banach–Mazur game" Rocky Mount. J. Math. , 17 (1987) pp. 227–276 |
[a11] | S.M. Ulam, "The Scottish Book" , A LASL monograph , Los Alamos Sci. Lab. (1977) (Edition: Second) |
Banach-Mazur game. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Banach-Mazur_game&oldid=22057