Suppose that $(\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})$ is a filtered probability space with totally ordered index set $T$. A stopping time is a random variable $\tau: \Omega \to T$ (e.g. in https://en.wikipedia.org/wiki/Stopping_time). In order to parse this requirement, we require that $T$ is endowed with a $\sigma$-algebra. In typical cases where $T$ is $\mathbb{N}$ or $[0, \infty)$, I can understand that the implied $\sigma$-algebra is the corresponding Borel $\sigma$-algebra, but I am wondering what the $\sigma$-algebra is supposed to be if $T$ is an arbitrary totally ordered set.
My hypothesis is that we could just endow $T$ with the order topology and then make use of the associated Borel $\sigma$-algebra. Is this correct? Do we have a choice of $\sigma$-algebras on $T$ for which $\tau$ is still considered a stopping time?