TITLE: A question on open mappings
QUESTION [5 upvotes]: Let $X$ be a metric space. $f: X \to Y=f(X)$ is open and continuous mapping.
Must $Y$ be a metric space? Thanks for your help.

REPLY [5 votes]: No. Let $X = \mathbb{R}$ and let $Y = \{a, b\}$ have the topology $\{\emptyset, \{a\}, \{a, b\}\}$.  Define the map $f: \mathbb{R} \to Y$ by $f(0) = b$ and $f(x) = a$ for all $x \in \mathbb{R} \setminus \{0\}$.  Then, $f$ is open, continuous, and onto, but $Y$ is not metrizable.