19.7.7 Properties of sets

Problem: Let \( A \subset \mathbb{R} \) be an arbitrary set, let \( \operatorname{Int} A \) is the set of its interior points. Prove the equivalence of the following three statements: 1) Int \( \bar{A}=\emptyset ; 2 \) ) the set \( A \) is nowhere dense; 3 ) the set \( \bar{A} \) is nowhere dense.