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Here is a definition on a paper named "Calculus on fractal subsets of real line"

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Definition 2 A subdivision $P_{[a,b]}$ (or just P) of the interval $[a,b], a

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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?

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    Please make your title relevant and also edit your post for errors in TeX and typing. – 2017-02-20
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    You are correct in both. I actually found the same problem with the word *refinement*, but you can see it as: if I add points, I can refine the approximation of my function – 2017-02-20
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    @ The Count I'm not an English speaker. May you be more precise about where are the errors? – 2017-02-20
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    @Giulio That's an useful viewpoint. Do you have an idea on how can be a "component" of P, something that is mostly made of parts not in P? – 2017-02-20
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    @nafagise as you see it, you can subdivide $[a,b]$ into $n$ components. In fact $[a,b]=\bigcup_{i=1}^{n-1} [x_i,x_{i+1}]$. Think to a pie and $n$ slices. Pie is $[a,b]$, a slice is $[x_i,x_{i+1}]$ – 2017-02-20
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    @ Giulio But then the components are components of $[a,b]$, not of P as defined, which is only a set of points. – 2017-02-20
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    As in my answer, this is a purely formal definition. it just makes writing a subdivision easier, but really you should think of the corresponding subintervals as being the subdivision being described. In this way, the definitions of component and refinement make sense. – 2017-02-20
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    There isn't a real difference between what you said and what I said. You can identify $[a,b]$ with its partition $P$. – 2017-02-20

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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

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.

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    If $\hat{P}$ were the definition, it would make full sense, but the author made an effort to define P as a **finite** set of **points**, instead of intervals. – 2017-02-20
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    It's purely formal. It just makes writing things easier. Use his definition in proofs, but use the above definition for intuition. – 2017-02-20
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    There is a tendency in mathematics to try and make your definitions require the least amount of information that is necessary, while still being well defined. That's what's happening here. – 2017-02-20