The **Choquet game** is a topological game named after Gustave Choquet, who was in 1969 the first to investigate such games.^{ [1] } A closely related game is known as the **strong Choquet game**.

Let be a non-empty topological space. The Choquet game of , , is defined as follows: Player I chooses , a non-empty open subset of , then Player II chooses , a non-empty open subset of , then Player I chooses , a non-empty open subset of , etc. The players continue this process, constructing a sequence If then Player I wins, otherwise Player II wins.

It was proved by John C. Oxtoby that a non-empty topological space is a Baire space if and only if Player I has no winning strategy. A nonempty topological space in which Player II has a winning strategy is called a **Choquet space**. (Note that it is possible that neither player has a winning strategy.) Thus every Choquet space is Baire. On the other hand, there are Baire spaces (even separable metrizable ones) which are not Choquet spaces, so the converse fails.

The strong Choquet game of , , is defined similarly, except that Player I chooses , then Player II chooses , then Player I chooses , etc, such that for all . A topological space in which Player II has a winning strategy for is called a **strong Choquet space**. Every strong Choquet space is a Choquet space, although the converse does not hold.

All nonempty complete metric spaces and compact T_{2} spaces are strong Choquet. (In the first case, Player II, given , chooses such that and . Then the sequence for all .) Any subset of a strong Choquet space which is a set is strong Choquet. Metrizable spaces are completely metrizable if and only if they are strong Choquet.^{ [2] }^{ [3] }

In mathematical analysis, a metric space *M* is called **complete** if every Cauchy sequence of points in *M* has a limit that is also in *M* or, alternatively, if every Cauchy sequence in *M* converges in *M*.

In mathematics, a **directed set** is a nonempty set *A* together with a reflexive and transitive binary relation ≤, with the additional property that every pair of elements has an upper bound. In other words, for any *a* and *b* in *A* there must exist *c* in *A* with *a* ≤ *c* and *b* ≤ *c*.

This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric topology.

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 mathematics, a **Baire space** is a topological space such that every intersection of a countable collection of open dense sets in the space is also dense. Complete metric spaces and locally compact Hausdorff spaces are examples of Baire spaces according to the Baire category theorem. The spaces are named in honor of René-Louis Baire who introduced the concept.

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 the mathematical field of topology, a **G _{δ} set** is a subset of a topological space that is a countable intersection of open sets. The notation originated in Germany with

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 the mathematical discipline of general topology, a **Polish space** is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named because they were first extensively studied by Polish topologists and logicians—Sierpiński, Kuratowski, Tarski and others. However, Polish spaces are mostly studied today because they are the primary setting for descriptive set theory, including the study of Borel equivalence relations. Polish spaces are also a convenient setting for more advanced measure theory, in particular in probability theory.

In topology and related areas of mathematics, a **topological property** or **topological invariant** is a property of a topological space which is invariant under homeomorphisms. That is, a property of spaces is a topological property if whenever a space *X* possesses that property every space homeomorphic to *X* possesses that property. Informally, a topological property is a property of the space that can be expressed using open sets.

In topology, a branch of mathematics, a **retraction** is a continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace. The subspace is then called a **retract** of the original space. A **deformation retraction** is a mapping that captures the idea of *continuously shrinking* a space into a subspace.

A subset of a topological space has the **property of Baire**, or is called an **almost open** set, if it differs from an open set by a meager set; that is, if there is an open set such that is meager. Further, has the **Baire property in the restricted sense** if for every subset of the intersection has the Baire property relative to .

In descriptive set theory, within mathematics, **Wadge degrees** are levels of complexity for sets of reals. Sets are compared by continuous reductions. The **Wadge hierarchy** is the structure of Wadge degrees. These concepts are named after William W. Wadge.

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.

In descriptive set theory, the **Borel determinacy theorem** states that any Gale–Stewart game whose payoff set is a Borel set is determined, meaning that one of the two players will have a winning strategy for the game.

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.

In mathematics, a **Hurewicz space** is a topological space that satisfies a certain basic selection principle that generalizes σ-compactness. A Hurewicz space is a space in which for every sequence of open covers of the space there are finite sets such that every point of the space belongs to all but finitely many sets .

In mathematics, a **Rothberger space** is a topological space that satisfies a certain a basic selection principle. A Rothberger space is a space in which for every sequence of open covers of the space there are sets such that the family covers the space.

In mathematics, a **-space** is a topological space that satisfies a certain a basic selection principle. An infinite cover of a topological space is an -cover if every finite subset of this space is contained in some member of the cover, and the whole space is not a member the cover. A cover of a topological space is a -cover if every point of this space belongs to all but finitely many members of this cover. A **-space** is a space in which for every open -cover contains a -cover.

- ↑ Choquet, Gustave (1969).
*Lectures on Analysis: Integration and topological vector spaces*. W. A. Benjamin. ISBN 9780805369601. - ↑ Becker, Howard; Kechris, A. S. (1996).
*The Descriptive Set Theory of Polish Group Actions*. Cambridge University Press. p. 59. ISBN 9780521576055. - ↑ Kechris, Alexander (2012).
*Classical Descriptive Set Theory*. Springer Science & Business Media. pp. 43–45. ISBN 9781461241904.

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