TEN TOPOLOGIES FOR 1 x 7

Introduction THE study of topologies on X x Y is motivated by some outstanding deficiencies of the cartesian, that is the usual, topology on the product of spaces. (Throughout this paper all spaces will be assumed to be Hausdorff.) Firstly, the cartesian product of identification maps is not, in general, an identification map. As a consequence certain natural products such as the join and smash product which are formed as identifications of cartesian products, turn out to be non-associative [cf. (4)]. Further, the cartesian product of locally uncountable CW-complexes is not a CW-complex, and this is a difficulty in an important application of these complexes: to the singular complex of a space. Secondly, the cartesian topology does not behave well in considering maps from, rather than into, 1 x 7 . One example of this, the exponential law for function spaces, will be discussed in detail in a sequel to this paper (1) (the difficulty here is, however, not usually traced to the product topology). Another, and similar, example is the use made by Bourbaki in (2) of 'fonctions hypocontinues', which are simply bilinear functions X X Y-> Z, continuous in some topology on X x F other than the cartesian. Now there are indefinitely many natural product topologies [of. § 1]. There are also a surprising number which are relatively close to the cartesian product, have interesting and useful properties, and are in some respects better than the cartesian. Our main purpose is to prove the theorem: