Brown-Booth-Tillotson theory for classes of exponentiable spaces (Q837641)

Given a class \(\mathcal C\) of topological spaces, Brown, Booth and Tillotson introduced a \(\mathcal C\)-product for any pair \(X,Y\) of topological spaces. This \(\mathcal C\)-product is the set \(X\times Y\) with a topology which depends on \(\mathcal C\) and coincides with the usual topology on \(X\times Y\) if \(Y\) belongs to \(\mathcal C\). The \(\mathcal C\)-product of the spaces \(X\) and \(Y\) is denoted by \(X\times_{\mathcal C} Y\). If \(f:X\times Y \to Z\) is a function then letting \(e_f(x)(y)=f(x,y)\) for any \((x,y)\in X\times Y\), we obtain a function \(e_f\) from \(X\) to the set of functions from \(Y\) to \(Z\). The authors consider the product \(X\times_{\mathcal C} Y\) and show that the correspondence \(f\to e_f\) is a well-defined bijection if and only if \(\mathcal C\) is a subclass of the class of exponentiable spaces in the category of all topological spaces. Also, a necessary and sufficient condition is found for \(\mathcal C\) to guarantee that the correspondence \(f\to e_f\) is a homeomorphism.