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Compositions of Set Operations

  • D. W. Bressler (a1) and A. H. Cayford (a2)

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The set operations under consideration are Borel operations and Souslin's operation (). With respect to a given family of sets and in a setting free of any topological structure there are defined three Borel families (Definitions 3.1) and the family of Souslin sets (Definition 4.1). Conditions on an initial family are determined under which iteration of the Borel operations with Souslin's operation () on the initial family and the families successively produced results in a non-decreasing sequence of families of analytic sets (Theorem 5.2.1 and Definition 3.5). A classification of families of analytic sets with respect to an initial family of sets is indicated in a manner analogous to the familiar classification of Borel sets (Definition 5.3).

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References

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1. Bressler, D. W. and Sion, M., The current theory of analytic sets, Can. J. Math. 16 (1964), 207230.
2. Kuratowski, C., Les suites transfinies d'ensembles et les ensembles projectifs, Fund. Math. 28 (1937), 186195.
3. Lusin, N., Sur la classification de M. Baire, C. R. Acad. Sci. Paris 164 (1917), 9194.
4. Lusin, N., Sur les ensembles analytiques, Fund. Math. 10 (1927), 195.
5. Lusin, N. and Sierpinski, W., Sur quelques proprieties des ensembles (), Bull. Acad. Sci. Cracovie 1918, 3748.
6. Morse, A. P., The role of internal families in measure theory. Bull. Amer. Math. Soc. 50 (1944), 723728.
7. Sierpinski, W., Les ensembles boreliens abstraits, Ann. Soc. Polon. Math. 6 (1927), 5053.
8. Souslin, M., Sur une definition des ensembles mesurables B sans nombres transfinis, C. R. Acad. Sci. Paris 164 (1917), 8891.
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Compositions of Set Operations

  • D. W. Bressler (a1) and A. H. Cayford (a2)

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