Products with Closed Projections

Introduction. In this paper we study pairs (X, Y) of topological spaces which have the property that one or both of the projections from Xx Y is closed (maps closed sets to closed sets) or z-closed (maps zero sets to closed sets). These conditions on projections arise naturally from considerations involving the exponential map on function spaces ; in § 1 we make explicit their relationship to the exponential map by proving several "exponential laws." As applications of one of these, we give conditions necessary and sufficient that a product, or semiuniform product, of two fine spaces be fine. The second and third sections are devoted to a study of conditions under which projections will be closed or z-closed. The results of the third section, together with the characterization of fine products in the first section, give a new, and entirely topological, proof of Isbell's characterization of (finite) fine products, Theorems 32, 33, and 35 in Chapter VII of [6]. They also give a new characterization of the C*-pairs defined in [2]. I would like to thank A. W. Hager and W. W. Comfort for several helpful discussions concerning the results in this paper.