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A game that appeared in the famous Scottish Book [[#References|[a11]]], [[#References|[a6]]], where its initial version was formulated as Problem 43 by the Polish mathematician S. Mazur: Given the space of real numbers $\mathbf{R}$ and a non-empty subset $E$ of it, two players $A$ and $B$ play a game in the following way: $A$ starts by choosing a non-empty interval $I_{0}$ of $\mathbf{R}$ and then $B$ responds by choosing a non-empty subinterval $I _ { 1 }$ of $I_{0}$. Then player $A$ in turn selects a non-empty interval $I _ { 2 } \subset I _ { 1 }$ and $B$ continues by taking a non-empty subinterval $I_3$ of $I_2$. This procedure is iterated infinitely many times. The resulting infinite sequence of nested intervals $\{ I _ { n } \} _ { n = 0 } ^ { \infty }$ is called a play. By definition, the player $A$ wins this play if the intersection $\cap _ { n = 0 } ^ { \infty } I _ {n}$ has a common point with $E$. Otherwise $B$ wins. Mazur had observed two facts:

a) if the complement of $E$ in some interval of $\mathbf{R}$ is of the first Baire category in this interval (equivalently, if $E$ is residual in some interval of $\mathbf{R}$, cf. also [[Category of a set|Category of a set]]; [[Baire classes|Baire classes]]), then player $A$ has a winning strategy (see below for the definition); and

b) if $E$ itself is of the first Baire category in $\mathbf{R}$, then $B$ 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 [[#References|[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|topological space]] $X$ with $\emptyset \neq E \subset X$ and the two players $A$ and $B$ were choosing alternatively sets $W _ { 0 } \supset W _ { 1 } \supset \ldots$ from an a priori prescribed family of sets $\mathcal{W}$ which has the property that every element of $\mathcal{W}$ contains a non-empty open subset of $X$ and every non-empty open subset of $X$ contains an element of $\mathcal{W}$. As above, $A$ wins if $( \cap _ { n = 0 } ^ { \infty } W _ { n } ) \cap E \neq \emptyset$, otherwise $B$ wins. Oxtoby's theorem says that $B$ has a winning strategy if and only if $E$ is of the first Baire category in $X$; also, if $X$ is a [[Complete metric space|complete metric space]], then $A$ has a winning strategy exactly when $E$ is residual in some non-empty open subset of $X$.

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 $X$, two players (usually called $\alpha$ and $\beta$) alternatively choose non-empty open sets $U _ { 1 } \supset V _ { 1 } \supset U _ { 2 } \supset V _ { 2 } \supset \ldots$ (in this sequence the $U _ { n }$ are the choices of $\beta$ and the $V _ { n }$ are the choices of $\alpha$; thus, it is player $\beta$ who starts this game). Player $\alpha$ wins the play $\{ U _ { n } , V _ { n } \} _ { n = 1 } ^ { \infty }$ if $\cap _ { n = 1 } ^ { \infty } U _ { n } = \cap _ { n = 1 } ^ { \infty } V _ { n } \neq \emptyset$, otherwise $\beta$ wins. To be completely consistent with the general scheme described above, one may think that $E = X$ and $\alpha$ starts by always choosing the whole space $X$. This game is often denoted by $\operatorname{BM} ( X )$.

A strategy $s$ for the player $\alpha$ is a mapping which assigns to every finite sequence $( U _ { 1 } \supset V _ { 1 } \supset \ldots \supset U _ { n } )$ of legal moves in $\operatorname{BM} ( X )$ a non-empty open subset $V _ { n }$ of $X$ included in the last move of $\beta$ (i.e. $V _ { n } \subset U _ { n }$). A stationary strategy (called also a tactics) for $\alpha$ is a strategy for this player which depends only on the last move of the opponent. A winning strategy (a stationary winning strategy) $s$ for $\alpha$ is a strategy such that $\alpha$ wins every play in which his/her moves are obtained by $s$. Similarly one defines the (winning) strategies for $\beta$.

A topological space $X$ is called weakly $\alpha$-favourable if $\alpha$ has a winning strategy in $\operatorname{BM} ( X )$, while it is termed $\alpha$-favourable if there is a stationary winning strategy for $\alpha$ in $\operatorname{BM} ( X )$. It can be derived from the work of Oxtoby [[#References|[a8]]] (see also [[#References|[a4]]], [[#References|[a7]]] and [[#References|[a9]]]) that the space $X$ is a [[Baire space|Baire space]] exactly when player $\beta$ does not have a winning strategy in $\operatorname{BM} ( X )$. Hence, every weakly $\alpha$-favourable space is a Baire space. In the class of metric spaces $X$, a metric space is weakly $\alpha$-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 $X$ that neither $\alpha$ nor $\beta$ has a winning strategy. For instance, the Bernstein set $X$ in the real line (cf. also [[Non-measurable set|Non-measurable set]]) is a Baire space which does not contain a dense completely metrizable subspace (consequently $X$ does not admit a winning strategy for either $\alpha$ or $\beta$).

The above characterization of weak $\alpha$-favourability for metric spaces has been extended for some non-metrizable spaces in [[#References|[a10]]].

A characterization of $\alpha$-favourability of a given [[Completely-regular space|completely-regular space]] $X$ can be obtained by means of the space $C ( X )$ of all continuous and bounded real-valued functions on $X$ equipped with the usual sup-norm $\| f \| _ { \infty } : = \operatorname { sup } \{ | f ( x ) | : x \in X \}$. The following statement holds [[#References|[a5]]]: The space $X$ is weakly $\alpha$-favourable if and only if the set

\begin{equation*} \{ f \in C ( X ) : f \ \text{attains its maximum in} \ X \} \end{equation*}

is residual in $C ( X )$. In other words, $X$ is weakly $\alpha$-favourable if and only if almost-all (from the Baire category point of view) of the functions in $C ( X )$ attain their maximum in $X$. The rich interplay between $X$, $C ( X )$ and $\operatorname{BM} ( X )$ is excellently presented in [[#References|[a3]]].

The class of $\alpha$-favourable spaces (spaces which admit $\alpha$-winning tactics) is strictly narrower than the class of weakly $\alpha$-favourable spaces. G. Debs [[#References|[a2]]] has exhibited a completely-regular topological space $X$ which admits a winning strategy for $\alpha$ in $\operatorname{BM} ( X )$, but does not admit any $\alpha$-winning tactics in $\operatorname{BM} ( X )$. Under the name  "espaces tamisables" , the $\alpha$-favourable spaces were introduced and studied also by G. Choquet [[#References|[a1]]].

[[#References|[a10]]] is an excellent survey paper about topological games (including $\operatorname{BM} ( X )$).

====References====
<table>
<tr><td valign="top">[a1]</td> <td valign="top">  G. Choquet,   "Une classe régulières d'espaces de Baire"  ''C.R. Acad. Sci. Paris Sér. I'' , '''246'''  (1958)  pp. 218–220</td></tr>
<tr><td valign="top">[a2]</td> <td valign="top">  G. Debs,   "Strategies gagnantes dans certain jeux topologique"  ''Fundam. Math.'' , '''126'''  (1985)  pp. 93–105</td></tr>
<tr><td valign="top">[a3]</td> <td valign="top">  G. Debs,   J. Saint-Raymond,   "Topological games and optimization problems"  ''Mathematika'' , '''41'''  (1994)  pp. 117–132</td></tr>
<tr><td valign="top">[a4]</td> <td valign="top">  M.R. Krom,   "Infinite games and special Baire space extensions"  ''Pacific J. Math.'' , '''55''' :  2  (1974)  pp. 483–487</td></tr>
<tr><td valign="top">[a5]</td> <td valign="top">  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</td>
</tr><tr><td valign="top">[a6]</td> <td valign="top">  "The Scottish Book: Mathematics from the Scottish Café"  R.D. Mauldin (ed.) , Birkhäuser  (1981)</td></tr>
<tr><td valign="top">[a7]</td> <td valign="top">  R.A. McCoy,   "A Baire space extension"  ''Proc. Amer. Math. Soc.'' , '''33'''  (1972)  pp. 199–202</td></tr>
<tr><td valign="top">[a8]</td> <td valign="top">  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</td></tr>
<tr><td valign="top">[a9]</td> <td valign="top">  J. Saint-Raymond,   "Jeux topologiques et espaces de Namioka"  ''Proc. Amer. Math. Soc.'' , '''87'''  (1983)  pp. 499–504  {{MR|}} {{ZBL|0511.54007}} </td></tr>
<tr><td valign="top">[a10]</td> <td valign="top">  R. Telgárski,   "Topological games: On the fiftieth anniversary of the Banach–Mazur game"  ''Rocky Mount. J. Math.'' , '''17'''  (1987)  pp. 227–276  [[ZBL|0619.90110}}</td></tr>
<tr><td valign="top">[a11]</td> <td valign="top">  S.M. Ulam,   "The Scottish Book" , ''A LASL monograph'' , Los Alamos Sci. Lab.  (1977)  (Edition: Second)</td></tr>
</table>