Continuity of the cone functor

This paper looks at the continuity of a class of functors that includes as special cases the cone functor Γ and the suspension functor Σ. The purpose of the paper is to highlight a sufficient topological property satisfied by paracompact Hausdorff spaces, which guarantees the continuity. Since paracompact Hausdorff spaces constitute a large class of topological spaces studied in mathematics, we regard this as a strong result. petus for the present paper came from a certain confusion encountered in the book General Topology and Homotopy Theory by James [General Topology and Homotopy Theory, Springer-Verlag, 1984]. We give a counterexample showing that the cone functor is not continuous in the category of regular spaces, as stated in the book. Although the results in this paper concern functors, the emphasis of this paper is more on classical point set topology.