Given a real valued stochastic process $X_t,t\in [0,\infty)$ adapted to a filtration $F_t$, it will be quite tempting to claim that $$\tau(\omega):=\inf\{t\ge 0\mid X_t(\omega)\in B\}$$ in which $B$ is a Borel set is a stopping time with regard to $F_t$, because at first glance it seems to just tell you the moment at which the particle hits the set $B$ "for the first time". This is what I initially anticipated too. However, much to my surprise, in this post @Did gave a counterexample.
Nevertheless, in this Wikipedia article, it is claimed that when $X_t$ is the Brownian motion, $F_t$ is the filtration it generates and $B=(a,\infty)$, then $\tau$ is indeed a stopping time which corresponds to the stopping rule: "stop as soon as the Brownian motion exceeds the value $a$." It thus leads me to believe that there may exist some sufficient conditions on $X_t$, $F_t$, or $B$ under which $\tau$ is indeed a stopping time. Could anyone enlighten me further? Thanks.
EDIT
It might be worth mentioning that I am particularly interested in the following conditions:
1). $X_t$ is a (generalised) Itô process, meaning it comes from a multidimensional Brownian motion $(B_1,\cdots,B_m)$.
2). $F_t$ is the sigma algebra generated by that multidimensional Brownian motion, in particular, $F_t$ is generated by r.v.s of the form $B_1(s_1,\cdot),\cdots,B_n(s_n,\cdot);s_k\le t$. (By the way, Is this sufficient to make $F_t=F_t^+$?)