Proof That the Closure of the Complement of a Nowhere Dense Subset of a Topological Space is Dense in the Space

Theorem

Ifis a nowhere dense subset of a topological spaceandis the complement ofthenis dense in

Proof

Suppose on the contrary thatis not dense inThenexists and an open setsuch that

Hence

Hence

This is a contradiction becauseis nowhere dense insoandis dense in