Example

Let $a$ be any rational number. The mapping $x \to a + x$ of the rational line $\mathbf{Q}$ into itself is continuous on $\mathbf{Q}$, for the inverse image under this mapping of an open interval $]b, c[$ is the open interval

Likewise, the mapping $x \to ax$ is continuous on $\mathbf{Q}$; this is clear if $a = 0$, for then $ax = 0$ for all $x$; if $a \neq 0$ then the inverse image under this mapping of the open interval $]b, c[$ is the open interval with end-points $b/a$ and $c/a$.

2) The direct image of an open (resp. closed) set of $X$ under a continuous mapping $f : X \to X'$ is not necessarily open (resp. closed) in $X'$ (cf. § 5).