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Can we plot a path on any path-connected space such that it has the property of crossing from each point of the space only once?

I have to admit my knowledge on topology is quite limited so I am approaching this as intuitively as possible with little success. Perhaps this is some trivial result which I have simply not encountered yet.

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A path in $X$ is a continuous function from $p:[0,1]\to X$. The demand that we cross each point exactly once, is saying that $p$ is 1-1 and onto. So if $X$ is Hausdorff, then $p$ is then a homeomorphism, and so this is possible if and only if $X$ is homeomorphic to the unit interval itself. This means that $X$ must be compact, connected ,second countable, and have exactly two non-cut points (see here for some references).

An indiscrete space of size continuum is another trivial example where this is possible (to see we need some separation axiom on $X$).

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  • $\begingroup$ Thank you very much for this concise explanation. +1 I see I have a lot to study regarding the subject. $\endgroup$ – MathematicianByMistake Oct 22 '17 at 13:16
  • $\begingroup$ p:[0,1] -> R, x -> x is a path within R and R is not compact. $\endgroup$ – William Elliot Oct 22 '17 at 19:14
  • $\begingroup$ @WilliamElliot it doesn’t cross through all points of $\mathbb{R}$, though. $\endgroup$ – Henno Brandsma Oct 22 '17 at 19:15

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