A new approach to the Katětov-Tong theorem

Abstract

We give a new proof of the Katětov-Tong theorem. Our strategy is to first prove the theorem for compact Hausdorff spaces, and then extend it to all normal spaces. The key ingredient is how the ring of bounded continuous real-valued functions embeds in the ring of all bounded real-valued functions. In the compact case this embedding can be described by an appropriate statement, which we prove implies both the Katětov-Tong theorem and a version of the Stone-Weierstrass theorem. We then extend the Katětov-Tong theorem to all normal spaces by showing how to extend upper and lower semicontinuous real-valued functions to the Stone-\v Cech compactification so that the less than or equal relation between the functions is preserved.