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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 (the previous exercises amount to an outline of the proof). – t.b. Sep 4 '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 4 '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 4 '11 at 19:37
@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 4 '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 4 '11 at 20:08

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