# A question about path-connected and arcwise-connected spaces

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

## 2 Answers

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.

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: 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