Let $E$ be a completely metrizable separable topological space and $\mathscr E$ be its Borel $\sigma$-algebra. Consider a measurable map $F:E\to E$ such that
- if $f:E\to \mathbb R$ is continuous and bounded, then $f\circ F:E\to\mathbb R$ is continuous and bounded.
Let us say that the function $g$ is nice if:
$g$ is continuous and $\inf\limits_E g = 0$;
$\{g = 0\}:=\{x\in E: g(x) = 0\}$ is not empty;
there is $\delta>0$ such that for all $g(F(x))
whenever $g(x)<\delta$.
Claim 1: if $g$ is a nice function and $\{g = 0\}$ is compact, then $\rho(\cdot,\{g = 0\})$ is also a nice function for any metric $\rho$ on $E$ which agree with the given topology.
Unfortunately, I couldn't neither prove nor disprove this claim. I had an idea that if this claim can be disproved, it is sufficient to disprove a conequence of that:
Claim 2: if $A$ is a compact set s.t. $\rho(\cdot,A)$ is a nice function (i.e. it verifies the property 3.) then $\rho'(\cdot,A)$ is also nice whenever $\rho'\sim \rho$.
but for that one I also didn't manage to construct a counterexample. Any help is appreciated.