Effective Compactness and Sigma-Compactness

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).