Since the definitions for a Hausdorff space and for a separated map is so similar, I wondered about the implications between them.
Suppose $X$ is a Hausdorff space, and we have a map $f:X \rightarrow Y$ to some topological space $Y$. So assuming that for two $x_1 \neq x_2 \in X$ we have that $f(x_1) = f(x_2)$, we know from the definition of a Hausdorff space that $f$ is separated.
Now onto the converse. Assuming that $f$ is separated. The Hausdorff condition on $X$ holds for all $x_1 \neq x_2 \in X$ when $f(x_1) = f(x_2)$, but since it's dependent on $f(x_1) = f(x_2)$ we cannot guarantee that the condition holds for all points $x_1 \neq x_2 \in X$. Is this correct?
Edit: Here's the definition of a separated map: A continuous map $f:X \rightarrow Y$ is called separated if any two points $x_1 \neq x_2 \in X$ with $f(x_1) = f(x_2)$ can be separated in the sense that there exists open sets $U_1, U_2 \subseteq X$ such that $x_1 \in U_1$, $x_2 \in U_2$ and $U_1 \cap U_2 = \emptyset$