Compositions of Set Operations

Bressler, D. W. ; Cayford, A. H.

Canadian journal of mathematics, Tome 22 (1970) no. 2, pp. 227-234

Résumé

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).

Détail

DOI : 10.4153/CJM-1970-029-8

Bibliographie

[1] 1. Bressler, D. W. and Sion, M., The current theory of analytic sets, Can. J. Math. 16 (1964), 207–230. Google Scholar

[2] 2. Kuratowski, C., Les suites transfinies d'ensembles et les ensembles projectifs, Fund. Math. 28 (1937), 186–195. Google Scholar

[3] 3. Lusin, N., Sur la classification de M. Baire, C. R. Acad. Sci. Paris 164 (1917), 91–94. Google Scholar

[4] 4. Lusin, N., Sur les ensembles analytiques, Fund. Math. 10 (1927), 1–95. Google Scholar

[5] 5. Lusin, N. and Sierpinski, W., Sur quelques proprieties des ensembles (), Bull. Acad. Sci. Cracovie 1918, 37–48. Google Scholar

[6] 6. Morse, A. P., The role of internal families in measure theory. Bull. Amer. Math. Soc. 50 (1944), 723–728. Google Scholar

[7] 7. Sierpinski, W., Les ensembles boreliens abstraits, Ann. Soc. Polon. Math. 6 (1927), 50–53. Google Scholar

[8] 8. Souslin, M., Sur une definition des ensembles mesurables B sans nombres transfinis, C. R. Acad. Sci. Paris 164 (1917), 88–91. Google Scholar