The conditions for a player to win can involve notions like topological closure and convergence.
After more than 35 years, the term “topological game” became widespread, and appeared in several hundreds of publications.
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.
The game is defined by the target property and the allowed moves at each step.
Alternatively, the sequence of moves might have length some ordinal number other than ω.
Player I begins the game by picking a nonempty open subset
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: 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; computable 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 chromatic number of infinite graphs.
For a longer list and a more detailed account see the 1987 survey paper of Telgársky.