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Concept
Théorie descriptive des ensembles
Résumé
In mathematical logic, descriptive set theory (DST) is the study of certain classes of "well-behaved" subsets of the real line and other Polish spaces. As well as being one of the primary areas of research in set theory, it has applications to other areas of mathematics such as functional analysis, ergodic theory, the study of operator algebras and group actions, and mathematical logic.
Descriptive set theory begins with the study of Polish spaces and their Borel sets.
A Polish space is a second-countable topological space that is metrizable with a complete metric. Heuristically, it is a complete separable metric space whose metric has been "forgotten". Examples include the real line , the Baire space , the Cantor space , and the Hilbert cube .
The class of Polish spaces has several universality properties, which show that there is no loss of generality in considering Polish spaces of certain restricted forms.
Every Polish space is homeomorphic to a Gδ subspace of the Hilbert cube, and every Gδ subspace of the Hilbert cube is Polish.
Every Polish space is obtained as a continuous image of Baire space; in fact every Polish space is the image of a continuous bijection defined on a closed subset of Baire space. Similarly, every compact Polish space is a continuous image of Cantor space.
Because of these universality properties, and because the Baire space has the convenient property that it is homeomorphic to , many results in descriptive set theory are proved in the context of Baire space alone.
The class of Borel sets of a topological space X consists of all sets in the smallest σ-algebra containing the open sets of X. This means that the Borel sets of X are the smallest collection of sets such that:
Every open subset of X is a Borel set.
If A is a Borel set, so is . That is, the class of Borel sets are closed under complementation.
If An is a Borel set for each natural number n, then the union is a Borel set. That is, the Borel sets are closed under countable unions.
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