Metrization theorems concerning relative compactness

Many of earlier and recent results obtained for p-spaces and their relatives can be extended by using a simple and natural concept (relative compactness) which was defined and investigated in an earlier paper of the author.

In the present paper extensions of recent metrization theorems concerning p-spaces are dealt with. A recent metrization theorem of J. Nagata is extended to relative compactness. Under the assumption of the continuum hypothesis A.V. Arhangel'skiǐ's problem asking whether a space, each subspace of which is a paracompact p-space, contains a dense metrizable subspace, is solved affirmatively (and for the generality of relative compactness). Some results concerning the behaviour of the first axiom of countability and its generalizations under relative compactness are also included.