Example of a function such that the property $ \forall U \in \tau, f^{-1}(f(U)) \in \tau$ doesn't hold.

Question

What continuous function $f$, set and topology $\tau$ could serve as a counter-example of the following property : $ \forall U \in \tau, f^{-1}(f(U)) \in \tau$ ?

Answer 1

Take $\mathbb{R}$ with its usual topology.

Take $f$ where $f(x) = (x-1)x(x+1)$.

$f((-1, 1)) = [-a, a]$ where $a$ is the local maximum ($\rightarrow a= \frac 2 {3\sqrt 3}$)
$f^{-1}(f((-1,1))) = [-b, b]$ where $b > 1$ and $f(b) = a$ ($\rightarrow b = \frac 2 {\sqrt 3})$

So although $(-1,1)$ is open, $f^{-1}(f((-1,1)))$ is not.

Addendum: from $\mathbb{R}$ to $\mathbb{R}$, continuous open mappings (functions that transform any open set into an open set) are the monotonic functions. So a sufficient condition for your property to hold on $\mathbb{R}$, is that $f$ is monotonic.