ABSTRACT
It is very well known that the category Top fails to have decently behaved function spaces; i.e., it fails to be cartesian closed, and so it is not convenient for several applications. Given any category a with finite products, an a-object Y is exponential in a if the functor · × Y : a → a https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072621/6e3dda93-0297-4e21-9483-9ad6f30dbbc7/content/eq2047.tif"/> has a right adjoint. a is cartesian closed provided each of its objects is exponential in a [6]. In particular, an arbitrary space Y is exponential in Top if for every space Z there is a (unique) topology t on the set C (Y,Z) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072621/6e3dda93-0297-4e21-9483-9ad6f30dbbc7/content/eq2048.tif"/> of continuous functions on Y to Z, such that for every space X the exponential function E XYZ : C (X×Y,Z) → C (X, C t (Y,Z)), https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072621/6e3dda93-0297-4e21-9483-9ad6f30dbbc7/content/eq2049.tif"/> where E XYZ (f)(x)(y) = f ^ (x)(y) = f ^ (x,y) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072621/6e3dda93-0297-4e21-9483-9ad6f30dbbc7/content/eq2050.tif"/> for all x and y, is a bijection.
