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