Consider a real stochastic process $X=(X_t)_{t\geq 0}$ with continuous sample paths on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ together with an open set $U\subseteq\mathbb{R}$.
Setting $\Omega_s:=X_s^{-1}(U)$, does it hold that $\lim_{s\rightarrow t}\mathbb{P}(\Omega_s\,\Delta\,\Omega_t)=0$ for each $t\geq 0$?
I think this is true, but maybe you know some slick way of (dis)proving it?
(Here, $A\,\Delta\, B:= (A\cup B)\setminus (A\cap B)$ denotes the symmetric difference of two sets $A,B$.)