Let Y be open closed in X d Prove that a subset U of Y is re

Let Y be open (closed) in (X, d). Prove that a subset U of Y is relatively open (relatively closed) in Y iff it is open (closed) in X.

Solution

if U is open in X

then U = U intersection Y

So U open in Y (open sets in Y are the intersection of open sets in X with Y)

if U closed , we do this for U complement

and U\'intersection Y is open in Y

and So U intersection Y is closed in Y