If $X$ is a Hausdorff topological space and it is path-connected, then it is arcwise-connected.
A question about path-connected and arcwise-connected spaces
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general-topology
2 Answers
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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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0The book 'Continuum Theory' by Sam Nadler contains a fairly expedited proof in chapter 8. – 2017-07-16