On relatively normal spaces, relatively regular spaces, and on relative property (a)

Spaces which have a certain relative topological property in every larger space from a certain class are investigated. It is proved that a regular (Tychonoff) space Y is normal in every larger regular (Tychonoff) space if and only if Y is Lindelof or normal almost compact. A functionally Hausdorff space Y is regular in every larger functionally Hausdorff space if and only if Y is compact. A Hausdorff (regular, Tychonoff) space Y is relatively (a) in every larger Hausdorff (regular, Tychonoff) X if and only if Y is compact. (C) 1999 Elsevier Science B.V. All rights reserved.