I have the following problem:
I need to prove or give a counterexample for the following claim: The image of a Hausdorff space under a continuous map is Hausdorff.
I claimed that this is wrong and I gave the following example. Take $$f:\Bbb{R}_d\rightarrow \Bbb{R}_t;\,\,\,x\mapsto x$$ where $d$ is the discrete topology and $t$ the trivial one. This map is clearly continuous and the domain is Hausdorff. But then $f(\Bbb{R})$ is not Hausdorff anymore in the trivial topology.
Now I'm not sure if this works as a counterexample. Could someone maybe take a look?
Thank you very much.
