So I'm looking at this proof, which is presented as a problem in Gamelin & Greene but I'm having some trouble understanding it.
http://www.math.ru.nl/~vangool/teach/top/sorgenfreyplaneisnotnormal.pdf
Part (a), (b) and (c) are straightforward enough. However I'm having trouble following his reasoning on (d). I understood how everything is defined, I just don't see how the Baire Category Theorem applies, which says that the intersection of an infinite sequence of dense subsets of a complete metric space is dense. Could someone please explain it further?