TITLE: Is a continuous mapping which is also open (closed) must be closed(open)?
QUESTION [5 upvotes]: As the topic, Is a continuous mapping which is also open (closed) must be closed (open)?

REPLY [11 votes]: A minimal example: Let $X=\{0,1\}$ with the topology $\{\emptyset,\{1\},X\}$ (i.e. the Sierpiński space). Then the constant map $f:X\to X$, $f\equiv 1$ is open but not closed, and $g\equiv 0$ is closed but not open.