Here is a definition on a paper named "Calculus on fractal subsets of real line"
Page 4:
Definition 2 A subdivision $P_{[a,b]}$ (or just P) of the interval $[a,b], a<b$ is a finite set of points $\{a=x_0,x_1,...,x_n=b \}$, $x_i<x_{i+1}$. Any interval of the form $[x_i,x_{i+1}]$ is called a component interval or just a component of the subdivision P. If Q is any subdivision of $[a,b]$ and $P \subset Q$, then we say that Q is a refinement of P. If $a=b$, then the set $\{a\}$ is the only subdivision of $[a,b]$.
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The first thing that confuses me is that it names a "component" of P to intervals between the points in P. Now, my intuitive view of a component is that it is a part or piece. Something "inside" P, that is completely contained in P. But the "component" as defined is not really contained in P. Most of it is not in P.
Then it goes again saying that $P \subset Q$ and that Q is is a refinement of P. Again I'm confused. My intuitive understanding of a "refinement", is something that is being made by extracting elements, so a "refinement" Q of P should be $Q \subset P$, not the other way.
This repeated counter intuitive definitions make me think that I'm missing something essential
Here is a drawing I made of how I visualize it: the interval $[a,b]$ in black, P in red, is a set of points inside the interval, and a "component" of P is the interval $[x_2,x_3]$, in green. Q is a "refinement" of P, despite having parts that are not in P, like being $Q=P \cup {\color{Blue} {bluepoint}}$
Is this the correct interpretation, or I'm inverting the concepts in some way, or those are just unfortunate choices of words for a definition?
[EDIT]If my interpretation is correct, then what is the logic in the words choice?
If it helps, you may want to use the definition that a subdivision of $[a,b]$ is a set of intervals $$\hat{P}=\{[x_i,x_{i+1}] \mid 0\leq i \leq n-1,\: x_i<x_{i+1}, x_0=a, x_n=b\}$$
There is obviously a one-to-one correspondence between this definition $\hat{P}$ of subdivision and the definition $P$ of subdivision that you have given, but it is much longer. In this case, the containments are in a sense written 'dually'. So, if $P\subset Q$ in your notation, then that means for all $[x,y]\in \hat{P}$, there exist $n,m$ such that $[x,y]=\bigcup_{i=n}^{m-1} [x_i,x_{i+1}]$ where $[x_i,x_{i+1}]\in \hat{Q}$, which corresponds to how you think of a refinement.