“Choquet game”的意思、由来-开放百科全书

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 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

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 T₂ 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]

References

1. ^{{cite book|last1=Choquet|first1=Gustave|title=Lectures on Analysis: Integration and topological vector spaces|date=1969|publisher=W. A. Benjamin|isbn=9780805369601|url=https://books.google.com/books?id=tVz_RQAACAAJ|language=en}}
2. ^{{cite book|last1=Becker|first1=Howard|last2=Kechris|first2=A. S.|title=The Descriptive Set Theory of Polish Group Actions|date=1996|publisher=Cambridge University Press|isbn=9780521576055|pages=59|url=https://books.google.com/books?id=L4Jf_ZRxqt8C&pg=PA59|language=en}}
3. ^{{cite book|last1=Kechris|first1=Alexander|title=Classical Descriptive Set Theory|date=2012|publisher=Springer Science & Business Media|isbn=9781461241904|pages=43-45|url=https://books.google.com/books?id=WR3SBwAAQBAJ&pg=PA43|language=en}}