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A topological game is an infinite positional game of perfect information played between two players on a topological space. Players choose objects with topological properties such as points, open sets, closed sets and open coverings. Time is generally discrete, but the plays may have transfinite lengths, and extensions to continuum time have been put forth. The conditions for a player to win can involve notions like topological closure and convergence.

It turns out that some fundamental topological constructions have a natural counterpart in topological games; examples of these are the Baire property, Baire spaces, completeness and convergence properties, separation properties, covering and base properties, continuous images, Suslin sets, and singular spaces. At the same time, some topological properties that arise naturally in topological games can be generalized beyond a game-theoretic context: by virtue of this duality, topological games have been widely used to describe new properties of topological spaces, and to put known properties under a different light.

The term topological game was first introduced by Berge, who defined the basic ideas and formalism in analogy with topological groups. A different meaning for topological game, the concept of “topological properties defined by games”, was introduced in the paper of Rastislav Telgársky, and later "spaces defined by topological games"; this approach is based on analogies with matrix games, differential games and statistical games, and defines and studies topological games within topology. After more than 35 years, the term “topological game” became widespread, and appeared in several hundreds of publications. The survey paper of Telgársky emphasizes the origin of topological games from the Banach-Mazur game.

There are two other meanings of topological games, but these are used less frequently.

• The term topological game introduced by Leon Petrosjan in the study of antagonistic pursuit-evasion games. The trajectories in these topological games are continuous in time.
• The games of Nash (the Hex games), the Milnor games (Y games), the Shapley games (projective plane games), and Gale's games (Bridg-It games) were called topological games by David Gale in his invited address [1979/80]. The number of moves in these games is always finite. The discovery or rediscovery of these topological games goes back to years 1948-49.

Definitions and notation

Many frameworks can be defined for infinite positional games of perfect information. Here we use the following.

• An infinite positional game between two players $P_1$ and $P_2$ is defined as a pair $G(X,\mathrm{\Phi })$ where $X$ is a topological space with cardinality $\left|X\right|$ and $\mathrm{\Phi }:X\mapsto \mathcal{𝒫}(X)$ is an upper (and/or lower) semicontinuous multivalued map assigning to each position $x\in X$ a set $\mathrm{\Phi }(x)\in \mathcal{𝒫}=\{Y|Y\subset X\}$ of legal positions having the Vietoris topology. We assume that $P_1$ makes the first move, and we say that $x$ is a terminal position if $\mathrm{\Phi }(x)=\mathrm{\varnothing }$.

• We set $[X{]}^{\kappa }=\{Y\subset X\left|\right|Y|=\kappa \}$, $[X{]}^{<\kappa }=\{Y\subset X\left|\right|Y|<\kappa \}$, and $[X{]}^{>\kappa }=\{Y\subset X\left|\right|Y|>\kappa \}$, where $\kappa$ is a cardinal number. Furthermore, we denote with $\overline{E}$ the closure of a subset $E$ of a topological space $X$, and we use $\mathcal{ℱ}(X)$ to denote the collection of all closed subsets of $X$.

• A play of the game is a sequence of type $\omega$, where $\omega =\{0,1,2,\cdots \}$. In many analyses the following quantities are useful: $2^{\omega }=\{0,1\}\times \{0,1\}\times \cdots$ (homeomorphic to the Cantor Discontinuum), and ${\omega }^{\omega }=\omega \times \omega \times \cdots$ (homeomorphic to the set of irrational numbers). The result of a play is either a win or a loss for each player.

• A strategy for player $P_i$ is a function defined over every legal finite sequence of moves of player $P_{-i}$. We denote with $P_i\uparrow G$ the fact that player $P_i$ has a winning strategy for $G$, and we say that $G$ is determined if either $P_1\uparrow G$ or $P_2\uparrow G$.

• Two games $G_1$ and $G_2$ are said to be equivalent if $(P_1\uparrow G_1\iff P_1\uparrow G_2)\wedge (P_2\uparrow G_1\iff P_2\uparrow G_2)$.

• A strategy for $P_i$ is stationary if it depends only on the last move of $P_{-i}$; a strategy is Markov if it depends both on the last move of $P_{-i}$ and on the ordinal number of the move.

• Many analyses focus on the set of real numbers, and denote with $R$ the real line, and with $J$ the closed unit interval.

The Sierpiński game

An illuminating example of the connections between game-theoretic notions and topological properties is the Sierpiński game. Let $\mathfrak{E}$ be a family of subsets of a space $Y$ such that the following properties hold.

• $(E\in \mathfrak{E})\wedge (E\subset E^{\prime }\subset Y)\Rightarrow (E^{\prime }\in \mathfrak{E})$;

• $({\cup }_{n<\omega }{E}_n\in \mathfrak{E})\Rightarrow (\mathrm{\exists }m<\omega |E_m\in \mathfrak{E})$.

Furthermore, let's associate with each decreasing sequence $(A_n|n<\omega )$ of subsets of $Y$, a family $H(A_n)$ of subsets of $Y$ such that:

• $(A\in H(A_n))\wedge (A\subset A^{\prime }\subset Y)\Rightarrow (A^{\prime }\in H(A_n))$;

• $(B_0,B_1,\cdots \in H(A_n))\Rightarrow ({\cap }_{n<\omega }{B}_n\in H(A_n))$.

Now, given a subset $X$ of $Y$, consider the following game $S(X,Y,\mathfrak{E},H)$: each player chooses alternatively elements $E_0,E_1,\cdots \in \mathfrak{E}$, enforcing that $X\supset E_0\supset E_1\supset \cdots$. Player $P_2$ wins when $X\in H(E_n|n<\omega )$. If $P_2\uparrow S(X,Y,\mathfrak{E},H)$, then $X$ is called a smooth set.

The Sierpiński game $S(X,Y)$ is a particular instance of this setup, in which $Y$ is Euclidean, $\mathfrak{E}=[X{]}^{>\omega }$, and $H(A_n)=\{A\subset X|A\supset ({\cap }_{n<\omega }{\overline{E}}_n)\}$. It turns out that this game has interesting correlations with many topological concepts.

• If $P_1\uparrow S(X,J)$, then $J\setminus X$ contains a copy of the Cantor Discontinuum; if $P_2\uparrow S(X,J)$, then $X$ contains a copy of the Cantor Discontinuum.

• If $X$ is analytic in $J$, then $P_2\uparrow S(X,J)$; the converse has not been proven yet.

• If $J\setminus X$ contains an analytic set that is not Borel-separated from $X$, then $P_1\uparrow S(X,J)$. This implies that, if $X$ is a coanalytic non-Borel subset of $J$, then $P_1\uparrow S(X,J)$.

• The family $\{X\subset J|P_2\uparrow S(X,J)\}$ is closed under the Suslin operation.

Other topological games

Some other notable topological games are:

• the Banach-Mazur game — the first infinite positional game of perfect information to have been studied;
• the Ulam game — a modification of the Banach-Mazur game;
• the Banach game — played on a subset of the real line;
• the Choquet game — related to siftable spaces;
• the point-open game — in which $P_1$ chooses points and $P_2$ chooses open neighborhoods of them.

Many more games have been introduced over the years, to study, among others: the Kuratowski coreduction principle; separation and reduction properties of sets in close projective classes; Luzin sieves; invariant descriptive set theory; Suslin sets; the closed graph theorem; webbed spaces; MP-spaces; the axiom of choice; recursive functions. Topological games have also been related to ideas in mathematical logic, model theory, infinitely-long formulas, infinite strings of alternating quantifiers, ultrafilters, partially ordered sets, and the coloring number of infinite graphs.

For a longer list and a more detailed account see the 1987 survey paper of Telgársky.

References

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