Let F be a closed subset of a metric space M and p∈M∖F . Show that there are two disjoint open sets G and H in M such that p∈G and H⊆F .
I solved well but I think is not the way ... We take a point y belonging to F. We have inf {d (p, y)}> 0 after p belongs to (M-F). F y belongs to a subset that is closed because its complement M is then opened out all the points are interior points F and then we can say that inf {d (p, y) / p belongs to (M-F)} and y belonging to F > 0. Then we can consider a good B (p, r) ta r = inf {d (p, y) / y belongs to F} / 3. Also, a ball B (y, r). We have that G is an open ball in M and H is also open ball in M, and we have the union of the two is an empty set.
