Arcwise and pathwise connectivity in space filling curves

Question

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?

Answer 1

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 $[0,1]$.