Exponential homeomorphisms in the category of topological spaces with base point (Q624416)

If we write \(Y^X\) for the space of continuous maps \(f:X\to Y\), suitably topologized, then the exponential law looks just like the law in arithmetic. It has long been known that one needed some assumptions, although the original cases in topology used the case where the domain map was a closed unit interval. In the paper [Mich. Math. J. 14, 133--152 (1967; Zbl 0145.43002)], \textit{N. E. Steenrod} looked at ways to generalize the early results so as to be useful in topology. The present authors in [Topology Appl. 156, No.~13, 2264--2283 (2009; Zbl 1179.54024)] studied topologies on functions spaces which permit functorial laws of exponentiation. The purpose of this paper is to extend these results to categories of spaces with base points. In this case, the product is replaced by a suitable smash product. The two basic theorems give conditions for the exponential functor to be a well-defined bijection or to be a homeomorphism (when the first two spaces are based [CW-complexes). The theorems are technical and require some grasp of the early literature.