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If $X$ is a Hausdorff topological space and it is path-connected, then it is arcwise-connected.

t.b.
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3 Answers3

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A path-connected Hausdorff space is arc-connected. I don't know (but would like to) any simple proofs of this claim. One way is to prove that every Peano (meaning compact, connected, locally connected and metrizable) space is arc-connected and then note that the image of a path in a Hausdorff space is Peano. The former part is not very easy but the latter part is. For the proofs see Chapter 31 of General Topology by Stephen Willard.

LostInMath
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It depends on your definition of arcwise-connectedness: in some books path-connected and arcwise-connected are the same. In other literature arcwise-connected is stronger since you require a continuous inverse. You can find more info here.

Dennis Gulko
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    Dennis: where did you find the information that Hausdorff is not enough; would you please provide a reference? @Dylan: I'm not so sure Wikipedia has this wrong. It is e.g. Exercise 6.3.12 (a) on page 376 of Engelking's General topology (the previous exercises amount to an outline of the proof). – t.b. Sep 04 '11 at 19:13
  • @Theo Thanks for the reference. I will try to look it up later, and perhaps add it to the Wikipedia article if everything checks out (such a thing should have a citation!). – Dylan Moreland Sep 04 '11 at 19:32
  • @Dylan: It seems that LostInMath provides a reference to Chapter 31 of Willard, which is probably better than reference to an exercise (the outline of LostInMath seems to match the outline given by Engelking). Yes, adding a good reference to Wikipedia would be a great thing to do, thanks in advance! – t.b. Sep 04 '11 at 19:37
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    @Theo: In my edition of Engelking it’s on p. 462. But Ch. 31 of Willard does give a complete proof (via the Hahn-Mazurkiewicz theorem). – Brian M. Scott Sep 04 '11 at 19:59
  • @Brian: Thanks a lot for the confirmation. I have the 1989 revised edition of Engelking that appeared in the Heldermann Verlag. I don't have a copy of Willard, but I'll have a look next time I'm in the library. – t.b. Sep 04 '11 at 20:08
  • @Theo: Mine’s the 1977 Polish edition of the English translation, which was apparently already revised and expanded from the original Polish. Yours probably has a better binding! – Brian M. Scott Sep 04 '11 at 20:15
  • @Brian: To be honest, I haven't seriously tested the binding so far, so I can't tell for sure... That's because I also have a scanned version of it and I mostly use the book for looking things up for quick confirmation, so the electronic version is more convenient and the physical version is in almost brand-new condition. – t.b. Sep 04 '11 at 20:19
  • @Brian: In Willard's book the result appears as a corollary of Hahn-Mazurkiewicz, but in fact only the easier direction of H-M (a continuous image of the unit interval in a Hausdorff space is Peano) is needed in the proof (besides arc-connectedness of Peano spaces which is Thm 31.2). – LostInMath Sep 04 '11 at 21:07
  • Dennis: In order for you to understand this comment thread: @Dylan pointed out in a deleted comment that Wikipedia states (w/o reference) that Hausdorff is enough for concluding that path-connectedness implies arc-connectedness (in the strong sense). I provided a reference and Brian confirmed that reference as well as the reference given by LostInMath. – t.b. Sep 04 '11 at 21:09
  • @Theo: I was wrong in my initial comment in the answer. Thank you and the others for the correction. – Dennis Gulko Sep 05 '11 at 06:06
  • The book 'Continuum Theory' by Sam Nadler contains a fairly expedited proof in chapter 8. – John Samples Jul 16 '17 at 01:15
  • see also https://math.stackexchange.com/q/857066 and its answer. – Tom Collinge Oct 09 '19 at 13:33
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I've seen the following article given as a reference for the statement in question:

R. Börger, How to make a path injective, Recent developments of general topology and its applications-International Conference in Memory of Felix Hausdorff (1868-1942), Berlin, March 22-28, pp. 57-59, 1992.

pavel
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