A general insertion theorem for uniform locales

A general insertion theorem due to Preiss and Vilimovský is extended to the category of locales. More precisely, given a preuniform structure on a locale we provide necessary and sufficient conditions for a pair f ≥ g of localic real functions to admit a uniformly continuous real function in-between. As corollaries, separation and extension results for uniform locales are proved. The proof of the main theorem relies heavily on (pre-)diameters in locales as a substitute for classical pseudometrics. On the way, several general properties concerning these (pre-)diameters are also shown.