Let $X$ be a continuous stochastic process with index set $\mathbb{R}_+$, and state space $\mathbb{R}$. Let $\mathcal{F}$ be a filtration generated by $X$. For fixed $b\geq0$, let $T$ be the time of the first entrance to $[b,\infty]$, that is, $$T = \inf \{t \in \mathbb{R}_+: X_t\geq b\}.$$ Show that $T$ is a stopping time of $\mathcal{F}$.
My attempt was to write ${T\leq t}$ in terms of $X_t$ but in that case I have an uncountable union. Any suggestions on how to get through this? Can I use rational numbers somehow?