In the Riemann integral section, my book says:
A Partition of $[a, b]$ is any finite subset $P$ having the form
$$P = \{a=t_0<t_1<\dots<t_n = b \}$$
I have never seen this notation before. Is a partition just a set of disjoint open/semiopen intervals, all of which are subsets of $[a, b]$? For example, is $\{[0, 2), (2, 6), (6, 10]\}$ a partition of $[0, 10]$?
This isn't quite accurate. The partition is the set of points $\{t_0, t_1, . . . , t_n\}$, and we think of this as representing the (not-quite-disjoint) intervals $[t_0, t_1], [t_1, t_2], . . . , [t_{n-1}, t_n]$.
The notation $$P=\{a=t_0<t_1<...<t_n=b\}$$ just means that $P$ is a set of $n+1$-many distinct elements, the least of which is $a$ and the greatest of which is $b$; it also fixes the notation that we label those points in order.