Definition 5.1.4 (Fine).label Let $P=\curl{x_j}^{n}_{j=0},Q=\curl{y_j}^{n}_{j=0}\in\scr{P}([a,b])$, then $Q$ is finer than $P$, denoted $P\leq Q$, if $\fall 0\leq j\leq m,\exists 0\leq k\leq n$ such that $x_{j}=y_{k}$. For any $P\leq Q\in \scr{P}([a,b])$ then

1. (1)

   $\scr{P}([a,b])$ or $\scr{P}_{t}([a,b])$ is equipped with $\leq$ is a upward-directed set

2. (2)

   If $P\leq Q$, then $\s(P)\geq \s(Q)$

3. (3)

   $\fall \e>0,\exists P\in\scr{P}([a,b])$ with $\s(P)<\e$

If $(P,c),(Q,d)\in \scr{P}_{t}([a,b])$ then $(Q,d)$ is finer than $(P,c)$ if $Q$ is finer than $P$