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We know that a space filling curve is not injective from Netto's theorem.

We know that a Peano space is a compact, connected, locally connected metric space.

Essentially in pathwise connectivity there is a continuous function from the interval $I$ into a space $X$.

And in arcwise connectivity there is a homeomorphic function from the interval $I$ into a space $X$.

We are told that every Peano space is arcwise connected.

The Hahn–Mazurkiewicz theorem says that a Hausdorff space $X$ is a continuous image of the unit interval $I$ if and only if it is a Peano space.

This Hahn–Mazurkiewicz theorem characterizes the continuous image of an interval $I$ which we know is not injective.

My question is when trying to characterize the continuous image of an interval, knowing that a space filling curve is not injective, why does this theorem contain the idea of a Peano space which is arcwise connected?

I found another theorem which says that a Hausdorff space is pathwise connected if and only if it is arcwise connected.

Does this mean that in a Hausdorff space, arcwise connectivity and pathwise connectivity become equivalent statements?

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  • $\begingroup$ "Netto's theorem"? I'm not familiar with this. We know it can't be injective because it then would be a continuous bijection from the closed interval to the square, hence a homeomorphism (since the closed interval is compact and the square is obviously Hausdorff), which is absurd. $\endgroup$ – Aloizio Macedo Jul 12 '16 at 16:18
  • $\begingroup$ @Aloizio Macedo: Netto is one of the mathematicians that tend to get mentioned in historical accounts of curves from the topological and classical point set theoretic viewpoint (as opposed to from an algebraic geometry viewpoint), such as this paper by Hans Sagan. $\endgroup$ – Dave L. Renfro Jul 12 '16 at 16:33
  • $\begingroup$ @AloizioMacedo It's mentioned here: web.stanford.edu/~kupers/spacefillingcurves.pdf $\endgroup$ – Henno Brandsma Jul 12 '16 at 16:33
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Indeed the statements are equivalent. So we have in particular that the following are equivalent for $X$:

  • $X$ is a compact, connected, locally connected metric space.
  • $X$ is a compact, connected, locally pathwise connected metric space.
  • $X$ is a compact, connected, locally arcwise connected metric space.
  • $X$ is a compact, path-connected, locally connected metric space.
  • $X$ is a compact, arcwise connected, locally connected metric space.
  • $X$ is the continuous Hausdorff image of $[0,1]$.
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