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I've stumbled upon a mention of Cousin's theorem in the context of Henstock–Kurzweil integral and got confused. I do not understand why this fact is called a theorem and what makes it any remarkable, when it seems to be a more or less trivial consequence of compactness (in the sens that every open cover has a finite subcover).

Here is the statement taken from Wikipedia (slightly simplified):

Let $\mathcal{C}$ be a full cover of $[a, b]$, that is, a collection of closed subintervals of $[a, b]$ with the property that for every $x\in[a, b]$, there exists a $\delta > 0$ so that $\mathcal{C}$ contain all subintervals of $[a, b]$ which contain $x$ and are of length smaller than $\delta$. Then there exists a partition $a = x_0 < x_1 <\dotsb < x_n = b$ of $[a, b]$ such that $[x_{i-1},x_{i}]\in{\mathcal{C}}$ for all $i$.

Is there some historical context that makes this obvious consequence of compactness deserve to be called a theorem? If so, why is this theorem still so often mentioned in the context of Henstock–Kurzweil integral nowadays instead of just referring to the compactness of the interval? Am I missing something?

Clearly, the interval $[a,b]$ is covered by the interiors (the interiors relative to $[a,b]$) of closed intervals in $\mathcal{C}$ such that all their closed subintervals containing their midpoint are also in $\mathcal{C}$, so there is a finite set $\mathcal{D}$ of such closed intervals in $\mathcal{C}$ that covers $[a,b]$. Is the hard part supposed to be to prove that if $[a,b]$ is covered by a finite set $\mathcal{D}$ of closed intervals, then there is a partition $a = x_0 < x_1 <\dotsb < x_n = b$ of $[a, b]$ such that each $[x_{i-1},x_{i}]$ is contained in an element of $\mathcal{D}$ and contains that element's midpoint?

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    $\begingroup$ Possibly helpful: Rethinking the elementary real analysis course by Brian S. Thomson (American Mathematical Monthly 114 #6, June-July 2007, pp. 469-490; at publisher and at JSTOR). $\endgroup$ Commented Oct 2, 2022 at 9:46
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    $\begingroup$ It is in a sense a simpler version of Heine-Borel's theorem; however what is remarkable about it is that through this humble result one can construct an integral that is even more general than that of Lebesgue through the simple idea of Riemann. $\endgroup$ Commented Oct 2, 2022 at 14:36
  • $\begingroup$ @OliverDíaz, i do not see at all how it is simpler than Heine-Borel's theorem. It looks otherwise to me. It is even hard to state concisely and precisely. After heaving read different versions of it, i was not sure to understand the statement. $\endgroup$ Commented Oct 2, 2022 at 14:43

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After re-reading the Wikipedia article and checking out the suggested reference to Rethinking the elementary real analysis course by Brian Thomson, it looks like to make Cousin's theorem look noteworthy, it may be enough to restate it as follows:

for every gauge $\delta$ on a closed bounded interval, there is a $\delta$-fine (finite) subdivision of that interval,

where a subdivision $a = x_0 < x_1 <\dotsb < x_n = b$ is called $\delta$-fine if and only if for each interval $[x_i, x_{i+1}]$ of the subdivision, there is $x\in [x_i, x_{i+1}]$ such that $[x_i, x_{i+1}]\subset (x -\delta(x), x +\delta(x))$.

When stated in this form, at least it takes less effort to read and grasp the meaning of the statement than to prove it, and its usefulness is more clear.

I should admit that there is a geometric or combinatorial component in this statement in addition to the purely topological compactness property. While trying to write down a proof, I got it wrong two times because was not careful about the order of points on the interval. One proof of Cousin's theorem can be decomposed into a straightforward application of the compactness (Heine-Borel's theorem) and the following geometrico-combinatorial lemma:

For every finite cover $\mathcal{U}$ of an interval $[a, b]$ by open intervals, there is a subdivision $a = x_0 < x_1 <\dotsb < x_n = b$ such that for each interval $[x_i, x_{i+1}]$ of the subdivision, there is an interval in $\mathcal{U}$ containing $[x_i, x_{i+1}]$ and with midpoint in $[x_i, x_{i+1}]$.

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