Clone segments in Top and in Unif

Abstract

Continuous and uniformly continuous maps of finite powers of metric spaces are investigated, e.g. for every 0 ≤ m ≤ n ≤ ∞, a metric space X is constructed such that the category of all continuous maps of the spaces X ⁰ = {ø}, X ¹ = X, X ² =X × X,…, X ^k and the category of all their uniformly continuous maps are:

$$\begin{gathered} equal{\text{ }}exactly{\text{ }}when{\text{ }}\kappa \leqslant m{\mkern 1mu} {\text{ }}and \hfill \\ isomorphic{\text{ }}exactly{\text{ }}when{\text{ }}\kappa {\mkern 1mu} \leqslant {\mkern 1mu} {\text{ }}n. \hfill \\ \end{gathered} $$