Does a quotient map p : X to Y where X is Hausdorff and Y is not exist?
Choose a quotient map , where denotes the real line in the K-topology and Y be the quotient space obtained from by collapsing the set K to a point.
is Hausdorff, but Y is not Hausdorff since p(K) and p(0) in Y cannot be separated by disjoint open sets.