Proof;
To prove that if A⊆B then Int(A)⊆Int(B), where A and B are subsets of a topological space (X, T), we will use the definition of interior of a set A and B.
 |
| Let A and B are subsets of a topological space (X, T).If A⊆B then Int(A)⊆Int(B). |
By definition
Int(A)⊆ A and Int(B)⊆B
Let A⊆B
⇒ Int(A)⊆A⊆B
⇒ Int(A)⊆B
⇒ Int(A)⊆Int(B)
∵ Int(A) is an open set which is contained in B but Int(B) is a largest open set which is contained in B.
Hence proved that If A and B are subsets of Topological Space (X, T) such that A⊆B then Int(A)⊆Int(B).