TITLE: Hausdorff spaces and separated maps.
QUESTION [5 upvotes]: 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 $U_1 \cap U_2 = \emptyset$

REPLY [4 votes]: The notion of separated map is the relative version of the notion of a Hausdorff space. By relative one means that it's a statement about the fibers of a map $f\colon X\rightarrow Y$. 
A continuous map $f\colon X\rightarrow Y$ is separated iff all fibers $f^{-1}(y)$ are Hausdorff topological spaces for the induced topology on $f^{-1}(y)$. In particular, $X$ is Hausdorff iff the continuous map from $X$ into the one-point topological space is separated.