# Proof;

To show that boundary of A is empty iff A is closed and open, we will use definition of closure and interior with focus on closed and open conditions,

cl(A)=A when A is closed and Int(A)=A when A is open.

Show that boundary of A is empty iff A is closed and open. |

Let boundary of a subset A of a topological space (X,T) is empty, i.e Fr(A)=Φ.

We know that the Relations of cl(A) and Int(A) with a subset A of X and Fr(A) are

cl(A)=A∪Fr(A) and Int(A)=A-Fr(A)

∵ we supposed that Fr(A)=Φ

⇒ cl(A)=A and Int(A)=A

⇒ A is closed and open

Conversely,

Let A is both closed and open subset of a topological space (X,T).

We know that

cl(A)=A∪Fr(A) and Int(A)=A-Fr(A)

Since A is both closed and open, So

cl(A)=A and Int(A)=A

⇒ A=A∪Fr(A) and A=A-Fr(A)

⇒ Fr(A)⊆A and A∩Fr(A)=Φ

⇒ Fr(A)=Φ

**Hence proved that Fr(A)=Φ iff A is closed and open.**