# Topological game

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,[1][2][3] 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,[4] and later "spaces defined by topological games";[5] 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[6] 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[7] 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.

## The Sierpiński game

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

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

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

## Other topological games

Some other notable topological games are:

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.[6]

## References

1. C. Berge, Topological games with perfect information. Contributions to the theory of games, vol. 3, 165–178. Annals of Mathematics Studies, no. 39. Princeton University Press, Princeton, N. J., 1957.
2. C. Berge, Théorie des jeux à n personnes, Mém. des Sc. Mat., Gauthier-Villars, Paris 1957.
3. A. R. Pears, On topological games, Proc. Cambridge Philos. Soc. 61 (1965), 165–171.
4. R. Telgársky, On topological properties defined by games, Topics in Topology (Proc. Colloq. Keszthely 1972), Colloq. Math. Soc. János Bolyai, Vol. 8, North-Holland, Amsterdam 1974, 617–624.
5. R. Telgársky, Spaces defined by topological games, Fund. Math. 88 (1975), 193–223.
6. R. Telgársky, Topological Games: On the 50th Anniversary of the Banach-Mazur Game, Rocky Mountain J. Math. 17 (1987), 227–276. [1] (3.2MB PDF)
7. L. A. Petrosjan, Topological games and their applications to pursuit problems. I. SIAM J. Control 10 (1972), 194–202.