Not points' but nowhere dense sets' I think:after you remove the boundaries of the memebrs of some countable base what is left is zero-dimenional and Polish and perfect and nowhere locally compact, hence homeomorphic to the space of irrationals. NB the plane minus a countable set and the line minus a countable set are not homeomorphic: the former is connectedm the latter is not.

You may want to replace `non-normal' by normal in the first line.

There's no need to credit anyone I think as it follows almost straight from the definitions. Maybe check this book (ams.org/mathscinet-getitem?mr=278261) to see if it isn't simply well known. (By the way, it suffices that $A\delta B$: either one of $A$ and $B$ meets both $X$ and $Y$ and we're done or $\lbrace A,B\rbrace=\lbrace X,Y\rbrace$, in which case we have $X\delta Y$ too.)