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