Geodesic
Effective Compactness and Sigma-Compactness
Matematičeskie zametki, Tome 91 (2012) no. 6, pp. 840-852.

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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/

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