A subspace of a Hausdorff space is Hausdorff.
Let X be Hausdorff and
Since Y is a subspace, where C is some open set in X.
Let
is the intersection of open sets so it is open. Therefore, where U and V are open sets in . Again since is open, U and V can be constructed in such a manner that .
Thus, the subspace is Hausdorff.
Correct?
Once again, not quite. You need to start with two arbitrary points and construct two disjoint neighborhoods of them. You know you can do this in , in other words there exists disjiont open sets in with , now how can you project these down to open subsets in which are disjoint and contain ?