ABSTRACT
A basic locally compact space has a neighborhood basis consisting of compact sets for each of its points. In general, a weakly locally compact space need not be basic locally compact, though two concepts coincide in T2- or regular spaces, in which case the term locally compact can be used instead without confusion. On the one hand, one can drop altogether any separation conditions on the domains and still obtain the desired results, at the reasonable cost of imposing a certain “weak regularity” assumption only on the range space. In view of several applications, one is concerned with conditions sufficient to make the exponential function EXYZ continuous, or an embedding, or a bijection, or even a homeomorphism with respect to the compact-open topology. There are two obvious ways to attack the natural question of how to obtain the standard results under different or relaxed separation assumptions.
