For a subset A of a metric space Xd Prove that A not equal X

For a subset A of a metric space (X,d). Prove that A not equal X) X in int A if and only if d(x,X-A) > 0 (Assume

A is a metric space of (X,d)

Let x belongs to int A

Then d(x,X-A) will be d(x,y) where y is not 0 as A not equals X

As x is in interior of A, x is not in boundary of A

Hence definite d(x,y) >0 thus d(x,X-A)>0

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Only if part:

Let d(x,X-A)>0

This implies x and X-A are different

i.e. x does not lie on the boundary of A

Hence x belongs to int A.