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Here is Professor Terry Tao's proof of the Jordan curve theorem using complex analysis, I more or less followed the proof until the following paragraph (see section 4). (Actually there is no need to read everything before the following paragraph to answer the question.)

https://terrytao.wordpress.com/2016/10/02/math-246a-notes-3-cauchys-theorem-and-its-consequences/

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enter image description here There are two things which I do not understand:

firstly, how do we know the boundary of $\Omega_\delta$ consists of one or more simple closed curves? How to rigorously demonstrate that its boundary cannot be some random collection of segments? Also, why does the fact that the union of squares is connected implies the boundary of $\Omega_\delta$ consists of exactly one simple closed curve. Also the sentence on why the boundary of $\Omega_\delta$ is simple got me confused.(last sentence of the paragraph above the picture)

secondly, what's the point of covering $N_{\epsilon/10}(\gamma([a,b]))$? Why not just cover $\gamma([a,b])$?

Any explanation would be immensely appreciated!!

Here $W$ denotes the winding number. $N_{\epsilon/10}(\gamma([a,b]))$ is the set of points $z\in\mathbb{C}$ such that $\text{distance}(z,\gamma)<\epsilon/10.$

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  • $\begingroup$ I think I have figured out my first question. But I still don't understand why it is necessary to cover $N_{\epsilon/10}(\gamma([a,b]))$. $\endgroup$
    – Simplyorange
    May 22, 2020 at 17:39
  • $\begingroup$ @Simplyorange- A cover of $N_{\epsilon/10}$ is also a cover of $\gamma$. So everything that Tao says about covering $N$ can be re-phrased as saying that we're taking covers of $\gamma$ by squares, and then assuming that the side lengths of those squares go to $0$. So essentially you're right, we could have gotten away with just talking about covering $\gamma$ also $\endgroup$
    – user67803
    May 26, 2020 at 17:30
  • $\begingroup$ It looks like you figured out the first question. Do you mind writing it out in an answer? You can answer your own first question and still leave the 2nd open for others. $\endgroup$
    – D.R.
    Nov 22, 2021 at 8:40

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The author first uses $10$ with $0\le\kappa<\frac1{10}$ after Exercise $61$.

$10$ also happens to be the number of line segments used to construct the polygon illustrated before Proposition $62$, which I guess are therefore related variables.

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