Let $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{\ge 0},P)$ be a filtered probability space. Let $X_t \in \mathbb{R}^n$ be a continuous stochastic process adapted to $\mathcal{F}_t$. Let $A \subset \mathbb{R}^n$. For instance $\tau$ is defined as
$$ \tau = \mathrm{inf}\{t\ge 0, X_t \in A \}. $$
Then, for showing that $\tau$ is a stopping time,
$$ \{\tau \le t\}= \cup_{s\in [0,t]\cap Q}\{X_s \in A\} $$
is considered. I am quite confused why the union is taken only over rational numbers? Some books say that it is due to continuity. What does it mean exactly? Does it mean the events $\{X_s \in A\}$ are not measurable over for some $s$ that are irrational numbers?
P.S.: I am from engineering background, just started reading on stochastic processes. It would be great if someone kindly clarify this.