Geodesic
Effective Compactness and Sigma-Compactness
Matematičeskie zametki, Tome 91 (2012) no. 6, pp. 840-852 Cet article a éte moissonné depuis la source Math-Net.Ru

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Using the Gandy–Harrington topology and other methods of effective descriptive set theory, we prove several theorems about compact and $\sigma$-compact sets. In particular, it is proved that any $\Delta_1^1$-set $A$ in the Baire space $\mathscr N$ either is an at most countable union of compact $\Delta_1^1$-sets (and hence is $\sigma$-compact) or contains a relatively closed subset homeomorphic to $\mathscr N$ (in this case, of course, $A$ cannot be $\sigma$-compact).
Keywords: effective descriptive set theory, effectively compact, $\sigma$-compact, the Baire space, Gandy–Harrington topology, $\Delta^1_1$-set. 
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V. G. Kanovei; V. A. Lyubetskii. Effective Compactness and Sigma-Compactness. Matematičeskie zametki, Tome 91 (2012) no. 6, pp. 840-852. http://geodesic.mathdoc.fr/item/MZM_2012_91_6_a4/

[1] Spravochnaya kniga po matematicheskoi logike. Chast II. Teoriya mnozhestv, red. K. Dzh. Baruaiz, Nauka, M., 1982 | MR | Zbl

[2] V. G. Kanovei, “Dobavlenie. Proektivnaya ierarkhiya Luzina: sovremennoe sostoyanie teorii”, Spravochnaya kniga po matematicheskoi logike. Chast II. Teoriya mnozhestv, ed. K. Dzh. Baruaiz, Nauka, M., 1982, 273–364

[3] V. G. Kanovei, “Razvitie deskriptivnoi teorii mnozhestv pod vliyaniem trudov N. N. Luzina”, UMN, 40:3 (1985), 117–155 | MR | Zbl

[4] V. G. Kanovei, V. A. Lyubetskii, “O nekotorykh klassicheskikh problemakh deskriptivnoi teorii mnozhestv”, UMN, 58:5 (2003), 3–88 | MR | Zbl

[5] V. G. Kanovei, V. A. Lyubetskii, Sovremennaya teoriya mnozhestv: borelevskie i proektivnye mnozhestva, MTsNMO, M., 2010

[6] V. Kanovei, Borel Equivalence Relations. Structure and Classification, Univ. Lecture Ser., 44, Amer. Math. Soc., Providence, RI, 2008 | MR | Zbl

[7] L. A. Harrington, A. S. Kechris, A. Louveau, “A Glimm–Effros dichotomy for Borel equivalence relations”, J. Amer. Math. Soc., 3:4 (1990), 903–928 | DOI | MR | Zbl

[8] G. Hjorth, “Actions by the classical {B}anach spaces”, J. Symbolic Logic, 65:1 (2000), 392–420 | DOI | MR | Zbl

[9] V. G. Kanovei, “Topologii, porozhdennye effektivno suslinskimi mnozhestvami, i ikh prilozheniya v deskriptivnoi teorii mnozhestv”, UMN, 51:3 (1996), 17–52 | MR | Zbl

[10] W. Hurewicz, “Relativ perfekte Teile von Punktmengen und Mengen $(A)$”, Fundam. Math., 12 (1928), 78–109 | Zbl

[11] J. Saint-Raymond, “Approximation des sous-ensembles analytiques par l'intérieur”, C. R. Acad. Sci. Paris Sér. A, 281:2-3 (1975), 85–87 | MR | Zbl

[12] A. S. Kechris, Classical Descriptive Set Theory, Grad. Texts in Math., 156, Springer-Verlag, New York, 1995 | MR | Zbl