SPACES WHICH ARE SCATTERED WITH RESPECT TO COLLECTIONS OF SETS

We study in this paper some of the topological structure associated with scattered spaces, a-scattered spaces, or, more generally, spaces which are cOllectionwise or a-collec tionwise scattered with respect to collections of sets which are either complete or satisfy some sort of generalized uni form first countability condition. A number of important counterexamples are spaces of this kind. One of the earliest results known to us of the type considered is that of Kura towski [K]: Every separable metric scattered space is an absolute Go. Telgarsky [T] proved that every T paracompact2 first countable scattered space is an absolute metric Go. The authors [WW] showed that every T first countable scattered2 l space has a A-base. (Both the results of Kuratowski and Telgarsky follow from this theorem which is a point of de parture for the present paper.) Subsequently they showe