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We give a new proof of the Kat\v{e}tov-Tong theorem. Our strategy is to firstprove the theorem for compact Hausdorff spaces, and then extend it to allnormal spaces. The key ingredient is how the ring of bounded continuousreal-valued functions embeds in the ring of all bounded real-valued functions.In the compact case this embedding can be described by an appropriatestatement, which we prove implies both the Kat\v{e}tov-Tong theorem and aversion of the Stone-Weierstrass theorem. We then extend the Kat\v{e}tov-Tongtheorem to all normal spaces by showing how to extend upper and lowersemicontinuous real-valued functions to the Stone-\v Cech compactification sothat the less than or equal relation between the functions is preserved.

A new approach to the Kat\v{e}tov-Tong theorem