Proof That For a Nowhere Dense Subset of a Metric Space and an Open Set U, An Open Subset of U exists With No Intersection With the Nowhere Dense Subset
For a nowhere dense subsetof a metric spaceand an open setan open subset ofexists with no intersection with
Letthe intersection ofwith the closure of
sinceis open andis dense in
Hence there exists
is open as the intersection of two open sets.
Henceexists such that