Let X be a topological space, and let Y subset of X have the subspace topology.
(a) If A is open in Y, and Y is open in X, show that A is open in X.
(b) If A is closed in Y, and Y is closed in X, show that A is closed in X.
(a) If A is open in Y, then $\displaystyle A=Y \cap U$ 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).