Consider the following subset of the real line:
in the subspace topology. In this space, the set [0,1] is open, since it is the intersection of an open set in R with Y. Similar (2,3) is open as a subset of Y; it is even open as a subset of R (I understand everything here). Since [0,1] and (2,3) are complements in Y of each other, we conclude that both [0,1] and (2,3) are closed as subsets of Y.
Y-[0,1] is open so [0,1] is closed.
Y-(2,3) is closed so why is (2,3) closed?