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I need hints to prove the implication “path-connected implies arc-connected” for metric spaces? [from Pugh chapter $2$ exercise $75$]

For simplicity, let me just call, given a path $f : [0,1] \to M$, a pair of points $(a,b)$ such that $f(a)=f(b)$ for distinct $a$ and $b$ an intersection point.

This exercise problem seems trivial enough if the number of intersection points is only finite. We can just remove all the inbetween image of intersection points $f( (a,b) )$ and get a homeomorphism. However, I cannot seem to tackle the case where there are an infinite number of intersection points.

I tried doing stuff like considering the supremum and infimum of the set of intersection points but that led nowhere because of some strange cases where the path is just very ugly/not convenient.

Pugh says this is only $1$ star difficulty which is why I feel like I’m just missing something. A hint in the right direction ( or even a complete proof ) would be very much appreciated. Thanks

Eric Wofsey
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Fernandeez nuts
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  • There is a related problem posted here but it was proved for Hausdroff spaces, something beyond the scope of the text I’m reading. (Or at least the chapter the exercise is on) – Fernandeez nuts Jun 26 '23 at 08:38
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    Welcome to MSE! In the future, try to type math with MathJax (pretty much the same as LaTeX if you're familiar with that one). – Bruno B Jun 26 '23 at 10:06
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    IMHO, this is an extremely difficult exercise. I do not see that the metric case should be considerably easier than the case of general Hausdorf topological spaces. – Jochen Jun 26 '23 at 12:16
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    Hi, please stop flagging my question because this is no duplicate. I try to be kind while typing, but, frankly, this is ridiculous. Anybody who has any knowledge at all regarding basic undergraduate calculus knows that arc-connectedness is a completely different concept, definitionally, to connectedness. Is the person flagging my question a high school kid or someone who has not enough respect to spare me to even bother reading my question. This is the second time I’ve been flagged and now my post is taken down. – Fernandeez nuts Jun 29 '23 at 17:45
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    @Fernandeeznuts: This is a very common occurrence, unfortunately. Most users here do not even bother to read questions when choosing to mark them as duplicates. – Eric Wofsey Jun 29 '23 at 18:09
  • Does this answer your question? A question about path-connected and arcwise-connected spaces – TheSilverDoe Jun 29 '23 at 18:33
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    @TheSilverDoe I have attempted to look at answers to this same question but in terms of Hausdorff spaces before; however, it is way beyond my realm of understanding. My question, as written, came from Pugh ( a very undergraduate-level analysis book) My question actually mostly comes from a place of trying to understand an undergraduate-version proof of this theorem ( without using complicated topology or measure theory concepts ). I raised this question because, as stated also, the question was rated 1 star (meaning average difficulty) by the author. – Fernandeez nuts Jun 29 '23 at 18:43
  • I'm not familiar with the particular book you are using but in most books exercises marked with a star are unusually difficult, not average difficulty. This exercise definitely is not of average difficulty for a basic real analysis text. – Eric Wofsey Jun 29 '23 at 18:49
  • Actually, a part of my question is also about knowing whether the case of metric spaces makes the proof easier as compared to Hausdorff spaces. I reasoned so because it was on an intro-to-analysis book exercise. @Jochen suggested not, what do you think? – Fernandeez nuts Jun 29 '23 at 18:50
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    The proof for normed space (easier) uses polygonal path I think, but I'm not sure it can be adapted since this concept needs an affine structure. I was not aware of this result since metric spaces are not necessarily Hausdorff... – julio_es_sui_glace Jun 29 '23 at 19:28
  • Oh yes big error from me, I have mistaken it for a weaker definition (Pseudometric)... dumb me – julio_es_sui_glace Jun 29 '23 at 19:32
  • Hi! Is this proof digestible without prior knowledge of subjects outside basic real analysis? If so, could you kindly link me to this proof? I seem to keep seeing the same proof using “peano continua”. – Fernandeez nuts Jun 30 '23 at 01:38

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