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I’m watching a lecture on the proof of the Cartan-Hadamard theorem, in which the lecturer claims that because $M$ is connected, it is path-connected.

I think that open subsets of locally path-connected spaces are path-connected. But I don’t know whether or why this result is true. Moreover, I suspect Riemannian manifolds are indeed locally path-connected, possibly due to the existence of normal neighbourhoods (in which in fact any two points are connected by a geodesic, if I am not mistaken or inaccurate).

Then, a manifold, being trivially open in itself, would be path-connected.

Any help would be appreciated.

EDIT:

I found this answer, but any help with the question/my lines of thoughts would still be helpful

Martin Geller
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1 Answers1

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Open, connected subsets of locally path-connected spaces are path-connected, because path components in locally path-connected spaces are open, and hence closed.

A manifold is locally path-connected because it is locally $\mathbb R^n$, and $\mathbb R^n$ is path-connected.

LSpice
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  • Thanks, do you have a link as to the proof of the first paragraph? I don’t understand why or how your “because … and closed” would be of relevance. Regarding your second paragraph, how would you explicitly construct a curve between two points lying in different charts? – Martin Geller Dec 01 '22 at 16:24
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    As @AnneBauval pointed out slightly after I posted my answer, you can find this in @‍Julien's answer to Showing that every connected open set in a locally path connected space is path connected, so it probably doesn't make a lot of sense for me to give a more detailed argument here. – LSpice Dec 01 '22 at 16:26
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    For your second question, I claim only local path-connectedness, so I need not join points in different charts. (But then, if the manifold is also connected, I can simply paste paths in a succession of overlapping charts; the proof that this works is simply the observation in my first paragraph.) – LSpice Dec 01 '22 at 16:28
  • Fair-enough, so you’ve just given a slicker full proof of global path connectedness than the other topological question you linked. Did suspect that my question was not a duplicate – Martin Geller Dec 01 '22 at 16:30
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    Re, my answer is the same as @‍Julien's and @‍muzzlator's (except that I didn't know about that question when I wrote my answer). I give about the same level of detail as @‍muzzlator, and slightly less than @‍Julien, but less detail should not be mistaken for slickness. I think that this question is a duplicate. – LSpice Dec 01 '22 at 16:33
  • I don’t think it’s the same, because your proof is manifold specific. Find a chart, find curves and paste together curves in charts. No resemblance whatsoever with the other answers – Martin Geller Dec 01 '22 at 16:36
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    @LSpice fyi: using atsign as you do in your above comments (and in your recent comment to me) does not notify the mentioned users since the scope of name resolution is local to the post (question or answer). Misusing it this way may confuse others into believing that it does notify - which may lead to important remarks being missed. – Bill Dubuque Nov 27 '23 at 18:18
  • @BillDubuque, re, you are quite right. Also I have abused at signs by using a zero-width joiner to allow two of them in one post. I consider these abuses justified to fight what I feel are warts in the syntax, but my opinion is idiosyncratic and you are right that they may also be generators of confusion. – LSpice Nov 27 '23 at 18:29
  • @LSpice I suppose you could also argue that it is in the hope that SE will someday broaden the scope. But even if that were true it would be redundant in your recent comment to me since I had already notified Dave via a comment on his answer. I confess that it bothers me that such distractions may inhibit comprehension of the (subtle) point I am making there - esp. for beginners (and it wasted my time - which is in short supply). – Bill Dubuque Nov 27 '23 at 18:52
  • @BillDubuque, re, I posted in chat, but, speaking of mis-using at-notifications, I almost certainly got it wrong. I do not think that this is likely to be a distraction for anyone other than you, but I also have no wish to cause harm or further distress, so I have deleted the comment on another post that bothered you. – LSpice Nov 27 '23 at 20:15