# A path on a path-connected space that crosses from each point only once.

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.

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 $X$).
• @WilliamElliot it doesn’t cross through all points of $\mathbb{R}$, though. – Henno Brandsma Oct 22 '17 at 19:15