A nonempty topological space where the first player has no winning strategy is the same as a Baire space, so in particular every Choquet space is a Baire space.
32.
A nonempty topological space where the first player has no winning strategy is the same as a Baire space, so in particular every Choquet space is a Baire space.
33.
A subset of Baire space has a corresponding subset of Cantor space under the map that takes each function from \ omega to \ omega to the characteristic function of its graph.
34.
This gives additional justification to the practice of restricting attention to Baire space and Cantor space, since these and any other Polish spaces are all isomorphic at the level of Borel sets.
35.
This leads to a representation of the Baire space as the set of all infinite paths passing through the full tree & omega; of finite sequences of natural numbers ordered by extension.
36.
Any connected T 1 space with more than one point is perfect . ( More interesting therefore are disconnected perfect spaces, especially totally disconnected perfect spaces like Cantor space and Baire space .)
37.
The arithmetical hierarchy can be defined on any effective Polish space; the definition is particularly simple for Cantor space and Baire space because they fit with the language of ordinary second-order arithmetic.
38.
In set theory, specifically descriptive set theory, the Baire space is used as a surrogate for the real numbers since the latter have some topological properties ( connectedness ) that are a technical inconvenience.
39.
General topologists use the term Baire space to refer to a broad class of topological spaces on which the notion of meagre set is not trivial ( in particular, the entire space is not meagre ).
40.
An alternative characterization, in the specific, important, case that X is Baire space ? ?, is that the analytic sets are precisely the projections of trees on \ omega \ times \ omega.
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