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This chapter may be viewed as a brief treatment of such parts of descriptive set theory as are needed in the main body of the text. The Borel hierarchy and analytic sets (Chapter 1) are developed further. The theorems of Souslin (analytic plus co-analytic imply Borel), Nikodym (preservation of the Baire property under the Souslin operation) and Marczewski (preservation of measurability under the Souslin operation) are stated (proved in more generality in Chapter 12). The Cantor Intersection Theorem is extended from closed (or compact) sets to analytic sets (Analytic Cantor Theorem). The Borel hierarchy is extended to the projective hierarchy: starting with the analytic sets $\sum^1_1$, their complements $\prod^1_1$ and the intersection of these, $\Delta^1_1$ (the Borel sets), one proceeds inductively: $\sum^1_{n+1}$ contains projections of $\prod^1_n$; their complements give $\prod^1_{n+1}$; intersections of these give $\Delta^1_{n+1}$, etc. The special importance of $\Delta^1_2$ is discussed.

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3 - Borel Sets, Analytic Sets and Beyond: Δ21

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3 - Borel Sets, Analytic Sets and Beyond: Δ21

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