On effective compactness and sigma-compactness

Using the Gandy – Harrington topology and other methods of effective descriptive set theory, we prove several theorems on compact and sigma-compact pointsets. In particular we show that any ${\mathrm{\Sigma }}_1^1$ set $A$ of the Baire space $N^N$ either is covered by a countable union of compact ${\mathrm{\Delta }}_1^1$ sets, or $A$ contains a subset closed in $N^N$ and homeomorphic to $N^N$ (and then $A$ is not covered by a sigma-compact set, of course).