# Upper semicontinuous function as a poinwise limit of continuous fuctions

The encyclopedia of mathematics claims, without proof, that an upper semicontinuous function on a completely regular topological space X is the pointwise limit of a decreasing sequence of continuous functions. I was able to find the proof (Bourbaki, General Topology, part II) only for the case when X is perfectly normal. Is the general statement above true, and if it is where can I find a proof?

## 1 Answer

See below Problem 1.7.15.c from “General topology” by Ryszard Engelking (Heldermann Verlag, Berlin, 1989).