Let $X,Y$ be topological spaces and $f : X \rightarrow Y$ be a continous,surjective map.
Then which of the following are true -
1) if $X$ is separable then $Y$ is separable.
2) if $X$ is first countable then $Y$ is first countable.
3) if $X$ is Hausdorff then $Y$ is Hausdorff.
4) if $X$ is regular then $Y$ is regular.
From the comments Topological properties preserved by continuous maps , and the wikipedia article,I think the first option is correct as then $f(X) = Y$.
Can we add more properties to the bag, if we add some other conditions in addition to surjectivity of $f$ ?