I am trying to prove the following statement regarding nowhere dense sets:
"In a metric space X, the frontier of an open set is the set of accumulation points of a discrete set."
As far as my attempts go, I have gone back and looked at an earlier proof of Baire's Category Theorem and worked out the details (as to my understanding, it should be similar). However, I am still not seeing how to prove the above using said theorem. Does anyone have any hints or suggestions as to how to proceed?
Note: I have already proven (as part of the above) that:
a)In a metric space X without isolated points, the closure of a discrete set in X is nowhere dense in X.
b) In any space X, the frontier of an open set is closed and nowhere dense.
c) Every closed nowhere dense set is the frontier of an open set.