  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 $\Sigma^1_1$ set $A$ of the Baire
space $N^N$ either is covered by a countable union of compact $\Delta^1_1$
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).
