Y is a subspace of X. If A is clopen in Y and Y is open in X then A is open in X. However, if A is closed in Y then A is closed in X. Correct?
Is there a case where A is closed in Y but open in X, no matter whether Y is open or closed or both in X?
If we consider the following subset of the real line
Y= [0,1] U (2,3)
in the subspace topology. The set [0,1] is open since it is the intersection o the open set (-1/2,3/2) of R with Y. Similarly, (2,3) is open as a subset of Y; it is open as a subset of R. Since [0,1] and (2,3) are complements in Y of each other, they are closed as subsets in Y.
Now, I need an example where A is closed in Y but open in the larger space of X.