Inmathematics, atopological game is an infinite game ofperfect information played between two players on atopological space. Players choose objects with topological properties such as points,open sets,closed sets andopen coverings. Time is generally discrete, but the plays may havetransfinite lengths, and extensions to continuum time have been put forth. The conditions for a player to win can involve notions liketopological closure andconvergence.

It turns out that some fundamental topological constructions have a natural counterpart in topological games; examples of these are theBaire 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 agame-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 withselection principles.

The termtopological game was first introduced byClaude Berge,^[1][2][3]who defined the basic ideas and formalism in analogy withtopological groups. A different meaning fortopological 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 theBanach–Mazur game.

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

• The termtopological game introduced byLeon Petrosjan^[7] in the study of antagonisticpursuit–evasion games. The trajectories in these topological games are continuous in time.
• The games ofNash (theHex games), theMilnor games (Y games), theShapley games (projective plane games), and Gale's games (Bridg-It games) were calledtopological games byDavid 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 infinitepositional games of perfect information.

The typical setup is a game between two players,I andII, who alternately pick subsets of a topological spaceX. In thenth round, playerI plays a subsetI_n ofX, and player II responds with a subsetJ_n. There is a round for every natural numbern, and after all rounds are played, playerI wins if the sequence

I₀,J₀,I₁,J₁,...

satisfies some property, and otherwise playerII wins.

The game is defined by the target property and the allowed moves at each step. For example, in theBanach–Mazur gameBM(X), the allowed moves are nonempty open subsets of the previous move, and playerI wins if${\displaystyle \bigcap _n{I}_n\ne \mathrm{\varnothing }}$.

This typical setup can be modified in various ways. For example, instead of being a subset ofX, each move might consist of a pair${\displaystyle (I,p)}$ where${\displaystyle I\subseteq X}$ and${\displaystyle p\in x}$. Alternatively, the sequence of moves might have length someordinal number other thanω.

• Aplay of the game is a sequence of legal moves

I₀,J₀,I₁,J₁,...

Theresult of a play is either a win or a loss for each player.

• Astrategy for playerP is a function defined over every legal finite sequence of moves ofP's opponent. For example, a strategy for playerI is a functions from sequences (J₀,J₁, ...,J_n) to subsets ofX. A game is said to be playedaccording to strategy s if every playerP move is the value ofs on the sequence of their opponent's prior moves. So ifs is a strategy for playerI, the play

${\displaystyle s(\lambda ),J_0,s(J_0),J_1,s(J_0,J_1),J_2,s(J_0,J_1,J_2),\dots }$

isaccording to strategy s. (Here λ denotes the empty sequence of moves.)

• A strategy for playerP is said to bewinning if for every play according to strategys results in a win for playerP, for any sequence of legal moves byP's opponent. If playerP has a winning strategy for gameG, this is denoted${\displaystyle P\uparrow G}$. If either player has a winning strategy forG, thenG is said to bedetermined. It follows from theaxiom of choice that there are non-determined topological games.
• A strategy forP isstationary if it depends only on the last move byP's opponent; a strategy isMarkov if it depends both on the last move of the opponentand 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.

LetY be a topological space, and letX be a subset ofY, called thewinning set. PlayerI begins the game by picking a nonempty open subset${\displaystyle I_0\subseteq Y}$, and playerII responds with a nonempty open subset${\displaystyle J_0\subseteq I_0}$. 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, andI wins if and only if

${\displaystyle X\cap \bigcap _{n\in \omega }{I}_n\ne \mathrm{\varnothing }.}$

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

• II has a winning strategy in the game if and only ifX is of thefirst category inY (a set is of thefirst category ormeagre if it is the countable union ofnowhere-dense sets).
• IfY is acomplete metric space, thenI has a winning strategy if and only ifX iscomeagre in some nonempty open subset ofY.
• IfX has theproperty of Baire inY, then the game is determined.

Some other notable topological games are:

• thebinary game introduced byUlam — a modification of the Banach–Mazur game;
• theBanach game — played on a subset of the real line;
• theChoquet game — related to siftable spaces;
• the point-open game — in which playerI choosespoints and playerII choosesopen neighborhoods of them;
• selection games — each round playerI chooses a (topological) collection andII chooses a member or finite subset of that collection. SeeSelection principle § Topological games.

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

For a longer list and a more detailed account see the 1987 survey paper of Telgársky.^[6]

• Topological puzzle

• Topological games
• General topology
• Game theory

• Articles with short description
• Short description matches Wikidata