Show that in a metric space, the union of a set and its exterior is dense everywhere

Show that in a metric space, the union of a set and its exterior is dense everywhere

Let X be a metric space and A ⊆ X. Since the exterior of A is Ext(A) = Int(X\A) = X\cl(A),

cl(A ∪ Ext(A)) = cl(A) ∪ cl(Ext(A)) = cl(A) ∪ cl(X\cl(A)) ⊇ cl(A) ∪ (X\cl(A)) = X.

Therefore, A ∪ Ext(A) is everywhere dense.