Q:

refinement of partition clarification

My book defines a refinement of a partition $P$ as such 
Given a partition $Ρ$ of the interval $[a, b]$, another partition $P^*$ of $[a, b]$ is called 
a refinement of $Ρ$ if each partition point of $Ρ$ is also a partition point of $P^*$. If $Ρ = 
\{x_0, ..., x_n\}$ and $P^*$ is a refinement of $P$, then for each index $i\geq 1$, the partition points  of $P^*$ that belong to the partition interval $[x_{i-1}, x_{i},]$ define a partition $P_i$, of the interval  $[x_{i-1}, x_i]$. 
What I don't get is what $P_i$ really means?  I have an example say that $P=\{[0,\frac{1}{2}],[\frac{1}{2},\frac{3}{4}],[\frac{3}{4},1]\}$ and $P^*=\{[0,\frac{1}{2}],[\frac{1}{2},\frac{3}{5}],[\frac{3}{5},\frac{3}{4}],[\frac{3}{4},1]\}$ where $P^*$ is a refinement of $P$. But I don't get how $P_i$ applies here? 

A:

$P_2$ is the partition of $[1/2,3/4]$ given by
$$
P_2=\Bigl\{\Bigl[\frac12,\frac35\Bigr],\Bigl[\frac35,\frac34\Bigr]\Bigr\}. 
$$
$P_1$ and $P_3$ consist of just an interval.