×

The exponential objects in TOP (a classical proof). (English) Zbl 0685.54011

For arbitrary spaces Y and Z let \(C_ t(Y,Z)\) denote the set C(Y,Z) of continuous maps on Y to Z equipped with some topology t. The topology t is said to be splitting on C(Y,Z) if for every space X and every function \(f\in C(X\times Y,Z)\) the adjoint function \(\hat f:\)X\(\to C_ t(Y,Z)\) such that \(\hat f(x)\)(y)\(\equiv f(x,y)\) is in \(C(X,C_ t(Y,Z))\). If for every X the continuity of \(\hat f\) implies that of f, then t is called jointly continuous on C(Y,Z). A space Y is exponential in TOP if for every space Z there is a splitting jointly continuous topology on C(Y,Z). A space Y is called corecompact if for every point \(y\in Y\) and every open set V containing y there is some open set W bounded in V and containing y (W is bounded in V if every open cover of V contains finitely many members covering W). The main result of the paper is the following theorem: a space Y is exponential in TOP if Y is corecompact.
Reviewer: V.K.Zakharov

MSC:

54C35 Function spaces in general topology
54C05 Continuous maps
PDFBibTeX XMLCite
Full Text: DOI