(a) If A is open in Y, then for an open set U in X by the definition of a subspace topology. An intersection of open sets is an open. Thus, A is open in X.
(b) is similar to (a).
(a) If A is open in Y, then for an open set U in X by the definition of a subspace topology. An intersection of open sets is an open. Thus, A is open in X.
(b) is similar to (a).