A **topological game** is an infinite 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.

- Basic setup for a topological game
- Definitions and notation
- The Banach–Mazur game
- Other topological games
- See also
- References

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. There are also close links with selection principles.

The term *topological game* was first introduced by Claude 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.

Many frameworks can be defined for infinite positional games of perfect information.

The typical setup is a game between two players, **I** and **II**, who alternately pick subsets of a topological space *X*. In the *n*th round, player **I** plays a subset *I*_{n} of *X*, and player II responds with a subset *J*_{n}. There is a round for every natural number *n*, and after all rounds are played, player **I** wins if the sequence

*I*_{0},*J*_{0},*I*_{1},*J*_{1},...

satisfies some property, and otherwise player **II** wins.

The game is defined by the target property and the allowed moves at each step. For example, in the Banach–Mazur game *BM*(*X*), the allowed moves are nonempty open subsets of the previous move, and player **I** wins if .

This typical setup can be modified in various ways. For example, instead of being a subset of *X*, each move might consist of a pair where and . Alternatively, the sequence of moves might have length some ordinal number other than ω_{1}.

- A
*play*of the game is a sequence of legal moves

*I*_{0},*J*_{0},*I*_{1},*J*_{1},...

- The
*result of a play*is either a win or a loss for each player.

- A
*strategy*for player**P**is a function defined over every legal finite sequence of moves of**P'**s opponent. For example, a strategy for player**I**is a function*s*from sequences (*J*_{0},*J*_{1}, ...,*J*_{n}) to subsets of*X*. A game is said to be played*according to strategy s*if every player**P**move is the value of*s*on the sequence of their opponent's prior moves. So if*s*is a strategy for player**I**, the play

- is
*according to strategy s*. (Here λ denotes the empty sequence of moves.)

- A strategy for player
**P**is said to be*winning*if for every play according to strategy*s*results in a win for player**P**, for any sequence of legal moves by**P'**s opponent. If player**P**has a winning strategy for game*G*, this is denoted . If either player has a winning strategy for*G*, then*G*is said to be*determined.*It follows from the axiom of choice that there are non-determined topological games. - A strategy for
**P**is*stationary*if it depends only on the last move by**P'**s opponent; a strategy isif it depends both on the last move of the opponent**Markov***and*on the ordinal number of the move.

The first topological game studied was the Banach–Mazur game, which is a motivating example of the connections between game-theoretic notions and topological properties.

Let *Y* be a topological space, and let *X* be a subset of *Y*, called the *winning set*. Player **I** begins the game by picking a nonempty open subset , and player **II** responds with a nonempty open subset . Play continues in this fashion, with players alternately picking a nonempty open subset of the previous play. After an infinite sequence of moves, one for each natural number, the game is finished, and **I** wins if and only if

The game-theoretic and topological connections demonstrated by the game include:

**II**has a winning strategy in the game if and only if*X*is of the*first category*in*Y*(a set is of the first category or meagre if it is the countable union of nowhere-dense sets).- If
*Y*is a complete metric space, then**I**has a winning strategy if and only if*X*is comeagre in some nonempty open subset of*Y*. - If
*X*has the Property of Baire in*Y*, then the game is determined.

Some other notable topological games are:

- the binary game introduced by Ulam — 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 player
**I**chooses points and player**II**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.^{ [6] }

In mathematics, more specifically in functional analysis, a **Banach space** is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well defined limit that is within the space.

In mathematics, more specifically in general topology, **compactness** is a property that generalizes the notion of a subset of Euclidean space being closed and bounded. Examples include a closed interval, a rectangle, or a finite set of points. This notion is defined for more general topological spaces than Euclidean space in various ways.

In mathematics, a **metric space** is a set together with a metric on the set. The metric is a function that defines a concept of *distance* between any two members of the set, which are usually called points. The metric satisfies a few simple properties. Informally:

In mathematics, a topological space is called **separable** if it contains a countable, dense subset; that is, there exists a sequence of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.

The **Baire category theorem** (BCT) is an important result in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space.

In mathematics, **general topology** is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is **point-set topology**.

In the area of mathematics known as functional analysis, a **reflexive space** is a Banach space that coincides with the continuous dual of its continuous dual space, both as linear space and as topological space. Reflexive Banach spaces are often characterized by their geometric properties.

In the mathematical fields of general topology and descriptive set theory, a **meagre set** is a set that, considered as a subset of a topological space, is in a precise sense small or negligible. The meagre subsets of a fixed space form a σ-ideal of subsets; that is, any subset of a meagre set is meagre, and the union of countably many meagre sets is meagre.

In functional analysis and related areas of mathematics, **Fréchet spaces**, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces. Fréchet spaces are locally convex spaces that are complete with respect to a translation-invariant metric. In contrast to Banach spaces, the metric need not arise from a norm.

In general topology, set theory and game theory, a **Banach–Mazur game** is a topological game played by two players, trying to pin down elements in a set (space). The concept of a Banach–Mazur game is closely related to the concept of Baire spaces. This game was the first infinite positional game of perfect information to be studied. It was introduced by Stanisław Mazur as problem 43 in the Scottish book, and Mazur's questions about it were answered by Banach.

In functional analysis and related branches of mathematics, the **Banach–Alaoglu theorem** states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology. A common proof identifies the unit ball with the weak* topology as a closed subset of a product of compact sets with the product topology. As a consequence of Tychonoff's theorem, this product, and hence the unit ball within, is compact.

In mathematics, the **axiom of determinacy** is a possible axiom for set theory introduced by Jan Mycielski and Hugo Steinhaus in 1962. It refers to certain two-person topological games of length ω. AD states that every game of a certain type is determined; that is, one of the two players has a winning strategy.

In functional analysis and related areas of mathematics, a **sequence space** is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field **K** of real or complex numbers. The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in **K**, and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are linear subspaces of this space. Sequence spaces are typically equipped with a norm, or at least the structure of a topological vector space.

**Determinacy** is a subfield of set theory, a branch of mathematics, that examines the conditions under which one or the other player of a game has a winning strategy, and the consequences of the existence of such strategies. Alternatively and similarly, "determinacy" is the property of a game whereby such a strategy exists.

In mathematical analysis, the **Kakutani fixed-point theorem** is a fixed-point theorem for set-valued functions. It provides sufficient conditions for a set-valued function defined on a convex, compact subset of a Euclidean space to have a fixed point, i.e. a point which is mapped to a set containing it. The Kakutani fixed point theorem is a generalization of Brouwer fixed point theorem. The Brouwer fixed point theorem is a fundamental result in topology which proves the existence of fixed points for continuous functions defined on compact, convex subsets of Euclidean spaces. Kakutani's theorem extends this to set-valued functions.

In topology and related areas of mathematics, a subset *A* of a topological space *X* is called **dense** if every point *x* in *X* either belongs to *A* or is a limit point of *A*; that is, the closure of A is constituting the whole set X. Informally, for every point in *X*, the point is either in *A* or arbitrarily "close" to a member of *A* — for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational number or has a rational number arbitrarily close to it.

The **Choquet game** is a topological game named after Gustave Choquet, who was in 1969 the first to investigate such games. A closely related game is known the **strong Choquet game**.

In mathematics, the **binary game** is a topological game introduced by Stanislaw Ulam in 1935 in an addendum to problem 43 of the Scottish book as a variation of the Banach–Mazur game.

In mathematics, the **Banach game** is a topological game introduced by Stefan Banach in 1935 in the second addendum to problem 43 of the Scottish book as a variation of the Banach–Mazur game.

In mathematics, a **selection principle** is a rule asserting the possibility of obtaining mathematically significant objects by selecting elements from given sequences of sets. The theory of **selection principles** studies these principles and their relations to other mathematical properties. Selection principles mainly describe covering properties, measure- and category-theoretic properties, and local properties in topological spaces, especially function spaces. Often, the characterization of a mathematical property using a selection principle is a nontrivial task leading to new insights on the characterized property.

- ↑ 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.
- ↑ C. Berge, Théorie des jeux à n personnes, Mém. des Sc. Mat., Gauthier-Villars, Paris 1957.
- ↑ A. R. Pears, On topological games, Proc. Cambridge Philos. Soc. 61 (1965), 165–171.
- ↑ 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.
- ↑ R. Telgársky, Spaces defined by topological games, Fund. Math. 88 (1975), 193–223.
- 1 2 R. Telgársky, "Topological Games: On the 50th Anniversary of the Banach-Mazur Game", Rocky Mountain J. Math. 17 (1987), 227–276.
- ↑ L. A. Petrosjan, Topological games and their applications to pursuit problems. I. SIAM J. Control 10 (1972), 194–202.

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