In mathematics, a subset of a topological space is called nowhere dense or rare if its closure has empty interior. In a very loose sense, it is a set whose elements are not tightly clustered anywhere. For example, the integers are nowhere dense among the reals, whereas an open ball is not.

How do you prove a set is nowhere dense?

A subset A ⊆ X is called nowhere dense in X if the interior of the closure of A is empty, i.e. (A)◦ = ∅. Otherwise put, A is nowhere dense iff it is contained in a closed set with empty interior. Passing to complements, we can say equivalently that A is nowhere dense iff its complement contains a dense open set (why?).

What is everywhere dense set?

A subset A of a topological space X is dense for which the closure is the entire space X (some authors use the terminology everywhere dense). A common alternative definition is: a set A which intersects every nonempty open subset of X.