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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?

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

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