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

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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.

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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.

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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](http://books.google.com/books?id=h4FsAAAAMAAJ)* (the previous exercises amount to an outline of the proof).2011-09-04
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    @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!).2011-09-04
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    @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!2011-09-04
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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).2011-09-04
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    @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.2011-09-04
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    @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!2011-09-04
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    @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.2011-09-04
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    @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).2011-09-04
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    Dennis: In order for you to understand this comment thread: @Dylan pointed out in a deleted comment that [Wikipedia](http://en.wikipedia.org/wiki/Connected_space#Arc_connectedness) 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.2011-09-04
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    @Theo: I was wrong in my initial comment in the answer. Thank you and the others for the correction.2011-09-05
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    The book 'Continuum Theory' by Sam Nadler contains a fairly expedited proof in chapter 8.2017-07-16