Definition 1 (wikipedia) A Suslin space is the image of a Polish space under a continuous mapping.
Definition 2 (Morris, Topology without Tears) A topological space (X, τ ) is said to be a Souslin space (or Suslin space) if it is Hausdorff and a continuous image of a Polish space.
Are these two definition equivalent? That is, can we prove (a) $f:X\to Y$ continuous, (b) $X$ Polish, imply $Y$ Hausdorff?
Note: Here we have two further definitions?!? Can someone reconcile the two definition of Suslin's condition?
If $X$ is Polish and $Y$ is $X$ with indiscrete topology then identity map is a continuous map from $X$ onto $Y$. Hence it is not true that a continuous image of a Polish space is Hausdorff.