Open Map

Definition (Open Map)

The function

$$f:(X,{\tau }_X)\to (Y,{\tau }_Y)$$

is an open map if $\forall$ open $U$ in $X$, $f(X)$ is open in $Y$. Clearly, for bijective $f$,

$$f^{-1}\text{ is continuous}⟺f\text{ is an open map}$$

From the previous example, clearly $[0,1/2)$ is an open set in the domain $[0,1)$. However, it’s image is not open in $S$.