I know that every metric space is a topological space. However, not all topological spaces are metric spaces, for example the cofinite topology is not metric since it is not Hausdorff.
But, I still have question which is whether every topological space is the continuous image of a metric space.
I am thinking about considering a discrete space in the domain to ensure that the function is continuous. But I could not finish the argument. Any help will be appreciated.