Because every Polish space is the continuous image of Baire space, it is often possible to prove results about arbitrary Polish spaces by showing that these properties hold for Baire space and are preserved by continuous functions.
42.
Because every Polish space is the continuous image of Baire space, it is often possible to prove results about arbitrary Polish spaces by showing that these properties hold for Baire space and are preserved by continuous functions.
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The choice of the Polish space Y in the third clause above is not very important; it could be replaced in the definition by a fixed uncountable Polish space, say Baire space or Cantor space or the real line.
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For any subset " T " of Baire space consisting only of characteristic functions of well orderings, there is an \ alpha such that each ordinal represented in " T " is less than \ alpha.
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The definition for a Baire space can then be stated as follows : a topological space " X " is a Baire space if every non-empty open set is of second category in " X ".
46.
The definition for a Baire space can then be stated as follows : a topological space " X " is a Baire space if every non-empty open set is of second category in " X ".
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To show that a subset of Cantor space or Baire space is analytic, you just check that the set is definable by a logical formula of the correct complexity, namely \ Sigma ^ 1 _ 1 in the analytical hierarchy.
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This restricted form applies in particular to the real line, the Baire space ? ?, the Cantor space 2 ?, and a separable Hilbert space such as " L " 2 ( " R " n ).
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A subset of Baire space is given the classification \ Sigma ^ 1 _ n, \ Pi ^ 1 _ n, or \ Delta ^ 1 _ n if and only if the corresponding subset of Cantor space has the same classification.
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It is much easier to prove this statement for Baire space ( set theory ), and that space is sufficiently similar to the real line to guide intuition . & mdash; Carl ( talk ) 14 : 03, 12 February 2009 ( UTC)
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