Understanding the definition of refinement of a partition $P$.

Question

Definition. Let $P$, $P'$ be partitions at $[a,b]$. We say that $P'$ is a refinement of $P$ if $P'$ $\supseteq$ $P$.

I did not understand really of the definition of refinement. Can you give me a example of $P'$, $P$ and $P'$ $\supseteq$ $P$?

Answer 1

$P'\supset P$ just means $P'$ contains all the points in $P$. It may also contain some others. Thus, suppose $P$ is this partition of $[0,1]$: $$ 0 < 0.1 < 0.3 < 0.7 < 1. $$ Then $P'$ would have to contain all of those same numbers. For example, $P'$ might be $$ \mathbf{0} < \mathbf{0.1} < 0.2 < \mathbf{0.3} < 0.5 < \mathbf{0.7} < 0.8 < 0.9 < 0.96 < \mathbf{1}. $$

Answer 2

For example if $[a,b] = [0,4]$ we may take $P = \{0,2,4\}$. Intuitively, this splits up the interval into two halves of equal length. If say $P' = \{0,1,2,4\}$ then this new partition is strictly finer because one of the two halves itself is partitioned.