# Showing these definitions of 'perfectly normal' spaces are equivalent

It seems to me that the first definition given is just strictly stronger than the second definition. I've tried to construct a function, using definition 2, that satisfies definition 1 but haven't made any noteworthy progress. Urysohn's lemma doesn't seem helpful here either.

Apparently, there's a really simple solution that only requires elementary facts about continuous functions but I haven't found anything useful over the span of multiple hours :P

Find a function $$f_1$$ for $$C$$, $$f_2$$ for $$D$$ and consider $$h=\frac{f_1}{f_1+f_2}$$. As $$f_1$$ and $$f_2$$ are never $$0$$ at the same time (and have values in $$[0,1]$$, so are $$\ge 0$$) this is well-defined, and continuous, as division is continuous on $$\Bbb R$$. Note that:
$$f^{-1}[\{1\}]= f_2^{-1}[\{0\}$$ and $$f^{-1}[\{0\}] = f_1^{-1}[\{0\}]$$ etc.