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

Theorem

For a nowhere dense subsetof a metric spaceand an open setan open subset ofexists with no intersection with

Proof

Letthe intersection ofwith the closure of

Thenand

sinceis open andis dense in

Hence there exists

is open as the intersection of two open sets.

Henceexists such that

Hence