This is an excerpt from the wikipedia page for Riemann integral which I don't understand. Can someone explain it in other words or graphically?
"Suppose that two partitions $P(x, t)$ and $Q(y, s)$ are both partitions of the interval $[a, b]$. We say that $Q(y, s)$ is a refinement of $P(x, t)$ if for each integer $i$, with $i ∈ [0, n]$, there exists an integer $r(i)$ such that $x_i = y_{r(i)}$ and such that $t_{i} = s_j$ for some $j$ with $j ∈ [r(i), r(i + 1))$."
To illustrate what does "refinement" mean, consider the interval $[0,10]$, and the partitions $P = (1,2, ..., 9)$, and $Q = (0.5, 1,2,...9)$, where the numbers represents the end points of the intervals in the partition.
Now $Q$ is a refinement of the partition $P$ because the intervals in the partition $Q$ are smaller, i.e contained, by the intervals in the partition $P$ because for every $i= 2,..., 9$ $$[i, i+1] \in Q \subseteq [i, i+1] \in P, (i = 2, ..., 9)$$, and $$[0,0.5] \subseteq [0,1] \in P \\ [0.5, 1] \subseteq [0,1] \in P$$
In general, we say that $Q$ is a refinement of $P$ if all the intervals in the partition $P$ are contained, as an interval, in the partition $Q$.