It's known that order topology is completely normal, so the lexicographic ordering on the unit square is also completely normal. It's also known that the lexicographic ordering on the unit square is not metrizable. I am interested in, is it perfectly normal (can any disjoint closed sets be separated by a continuous function)? And how to prove that.

It's known that order topology is completely normal, so the lexicographic ordering on the unit square is also completely normal. It's also known that the lexicographic ordering on the unit square is not metrizable. I am interested in whether it is perfectly normal. (A space is perfectly normal if for any two disjoint nonempty closed subsets, there is a continuous function $f$ to $[0,1]$ that "precisely separates" the two sets, meaning that the two closed sets are $f^{-1}(0)$ and $f^{-1}(1)$.) And how do we prove it?

Its known that order topology is completely normal so Lexiographic Ordering on the unit square is also completely normal. Its also known that Lexiographic Ordering on the unit square is not metrizable. I am interested in, is it perfectly normal (can any disjoint closed sets separate by continuous function)? And how to prove that.

It's known that order topology is completely normal, so the lexicographic ordering on the unit square is also completely normal. It's also known that the lexicographic ordering on the unit square is not metrizable. I am interested in, is it perfectly normal (can any disjoint closed sets be separated by a continuous function)? And how to prove that.