Definition
There are several equivalent definitions of analytic set. The following conditions on a subspace A of a Polish space are equivalent:
- A is analytic.
- A is empty or a continuous image of the Baire space ωω.
- A is a Suslin space, in other words A is the image of a Polish space under a continuous mapping.
- A is the continuous image of a Borel set in a Polish space.
- A is a Suslin set, the image of the Suslin operation.
- There is a Polish space and a Borel set such that is the projection of ; that is,
- A is the projection of a closed set in X times the Baire space.
- A is the projection of a Gδ set in X times the Cantor space.
An alternative characterization, in the specific, important, case that is Baire space, is that the analytic sets are precisely the projections of trees on . Similarly, the analytic subsets of Cantor space are precisely the projections of trees on .
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