Let $(\Omega,\mathscr F,(\mathscr F_t)_{t\geq 0},\mathsf P)$ be a complete filtered probability space and $X = (X_t)_{t\geq 0}$ be a cadlag stochastic process with value in a Polish space $E$. Is it true that for any closed subset $A$ of $E$ the first hitting time $ \tau = \inf\{t\geq 0:X_t\in A\} $ is a stopping time, i.e. $\{\tau\leq t\}\in \mathscr F_t$ for all $t\geq 0$; and what if $A$ is an open set.
As an example, we can consider $X$ with values in $\mathbb R$. It holds that the first hitting time of a closed half-line $ \tau = \inf\{t\geq 0:X_t\in[K,\infty)\} $ is a stopping time. Can we apply the following argument:
Let $\tau = \inf\{t\geq 0:X_t\in (K,\infty)\}$ then $ \{\tau\leq t\} = \bigcup\limits_{n=1}^\infty\{\tau_n\leq t\}\in \mathscr F_t $ where $\tau_n = \inf\{t\geq 0:X_t\in[K+1/n,\infty)\}$ and hence $\{\tau_n\leq t\}\in \mathscr F_t$.