The question's listed below:
Let f be a real-valued function defined on R (real). Show that the inverse image with respect to f of an open set is open, of a closed set is closed, and of a Borel set is Borel.
Thank you so much for the help!
The question's listed below:
Let f be a real-valued function defined on R (real). Show that the inverse image with respect to f of an open set is open, of a closed set is closed, and of a Borel set is Borel.
Thank you so much for the help!
That is not true unless your function is continuous. For example:
Let $\displaystyle f:\mathbb{R} \to \mathbb{R}$ define the function:
$\displaystyle f(x)=\left\{\begin{array}{lr}x & \text{if x is irrational}\\x+1 & \text{if x is rational}\end{array}\right.$
$\displaystyle f^{-1}(1,2) = ((0,1) \cap \mathbb{Q})\cup((1,2) \setminus \mathbb{Q})$. But this is the preimage of the open interval (1,2) whose preimage is neither open nor closed.