Relative collectionwise normal
by
Gary Grabner
Slippery Rock University, USA
Coauthors: Elise Grabner, Kazumi Miyazaki

For a subset A of a space X we say that A is collectionwise normal in X provided for every discrete collection of closed subsets of X, {F(a): a in A} there is a collection of open subsets of X, {U(a): a in A} discrete with respect to A such that for all a in A, the intersection of F(a) and A is contained in U(a). Suppose that A is a subset of a regular space X.

Theorem 1 If A is 1-paracompact in X then A is collectionwise normal in X.

Theorem 2 If A is strongly metacompact in X and collectionwise normal in X then A is weakly cp-paracompact in X.

Theorem 3 If A is collectionwise normal in X and developable in X then A is a metrizable subspace of X. In fact, A has an outerbase in X which is sigma discrete with respect to A.

Date received: January 31, 2001