A fast question (i dont want a proof):
Let and two topological spaces. Itīs not Hausdorff.
Let X_1, T_1)\to(X_2, T_2)" alt="fX_1, T_1)\to(X_2, T_2)" /> a function continuous. Its possible that to be Hausdorff?
I think no.
It is true however if you swap the spaces around i.e. f cts and X_2 not Hausdorff implies that X_1 is not Hausdorff.
Let be the real line with the standard topology; let be the product topological space , where is the with the indiscrete topology. Then is not a Hausdorff space but is a Hausdorff space. You can construct a surjective continuous map given , where and .
Consider an identity map between with the discrete topology and with the indiscrete or cofinite topology. Then the map is continuous and surjective, but the domain of the map is Hausdorff and the codomain of the map is non-Hausdorff.