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.