# example of a open function such that the restriction is not open

Give an example of a function $f:X \to Y$ and a subset $A \subset X$ such that f is open but $f_A$, the restriction of $f$ to $A$ is not open.

Can someone help me please? Thanks

• what do you mean by f is open? – RowanS Jun 21 '15 at 19:40
• @Rowan: It’s a standard notion: it means that if $U$ is open in the domain of $f$, then $f[U]$ is open in the codomain. – Brian M. Scott Jun 21 '15 at 19:44

HINT: Let $X=Y=\Bbb R$, take $f$ to be the identity function, and take $A$ to be a set that isn’t open in $X$.