Abstract
This paper is concerned with a portion of descriptive set theory, namely the theory of (boldface) \( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\sum }}_{n}^{1},{\text{ }}{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\prod }}_{n}^{1} \) and \( \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\Delta } _n^1 \) sets in Polish spaces, for n ≤ 3. We assume the reader has some familiarity with this subject, in fact with the logicians’ version of this subject. Moschovakis [37] is the basic reference and we generally follow his notation and terminology. A Polish space is a topological space homeomorphic to a separable complete metric space. All uncountable Polish spaces are Borel isomorphic, and a Borel isomorphism preserves \( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\sum }}_{n}^{1} \) sets, so as far as the abstract theory is concerned, there is only one space [37,1G]. But particular examples happen to live in particular spaces, so in this paper we will consider many different spaces, all Polish.
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Becker, H. (1992). Descriptive Set Theoretic Phenomena in Analysis and Topology. In: Judah, H., Just, W., Woodin, H. (eds) Set Theory of the Continuum. Mathematical Sciences Research Institute Publications, vol 26. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9754-0_1
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