Any proof (in English) that Generalised Peano Continua are continuous images of the real line?

Question

By generalised Peano continuum I mean a metric, connected, locally connected, locally compact space. I found a paper called A Hahn-Mazurkiewicz Theorem for generalized Peano continua, which says that this result was proved by Mazurkiewicz in his seminal paper Sur les lignes de Jordan. However it is a more than 40 pages long document in French and I don´t understand french, is there a more recent book in English that deals with this topic? I haven´t found any.

Answer 1

After a few days, I was able to find a way to prove the result. So I thought I should write an answer. The proof I wrote for my thesis is highly analogous to the section 31 on Peano Spaces of Willard´s General Topology. This answer can´t be understood without previously reading that section. The proof took me about two pages by latex with wide margins, and it is in spanish, so I´ll just sum up the steps here. Either way I´ll add the proof in Spanish in two images at the end of the answer.

You begin by proving the analogous to theorem 31.2:

If $X$ is a Generalised Peano Space and $U$ is an open connected subset of $X$ such that $\overline{U}$ is compact, then $U$ is path connected.

The proof is can be done the same way as the one of 31.2, thanks to the fact that $U$ is connected and locally connected and $\overline{U}$ is compact, so the sets $C_n$ of that proof will also be compact. By the way, if you try to understand it in detail, you´ll see that at the very least the proof of 31.2 in Willard´s General Topology is incomplete. More info about this proof can be found in this article. The proof can be fixed, although it´s a tedious task. May be worth it to use proofs from other books.

Anyways, with that lemma we know that Generalised Peano spaces are locally path connected, and thus are path connected. The next result you need is this one:

Generalised Peano spaces are $\sigma$-compact (that is, a countable union of compact subspaces).

To prove it you just have use that Generalised Peano spaces are metric, so they are paracompact, and that connected, locally compact, paracompact spaces are $\sigma$-compact (this last fact is proved in the second page of Appendix A of Spivak´s 'A Comprehensive Introduction To Differential Geometry. Vol1').

In Willard General Topology´s section about Peano spaces, they prove that Peano spaces are unformly locally arcwise connected. However, the same can´t be said of generalised Peano spaces. However, it will be enough to prove this weaker property:

If $X$ is a generalised Peano space and $C$ is a compact subspace of $X$, then $C$ is uniformly locally arcwise connected respect to $X$, that is, given $\varepsilon>0$ there is $\delta>0$ such that, if $a$ and $b$ are points of $C$ and $d(a,b)<\delta$, then there is an arc in $X$ from $a$ to $b$ with a diameter $<\varepsilon$.

The proof is essentially the same as Willard´s proof that Peano spaces are uniformly locally arcwise connected.

After this, you are prepared to prove that any Generalised Peano Space $X$ is a continuous image of $\mathbb{R}$. The proof is analogous to Willard´s proof of 31.5. In that proof, he begins by defining a surjective function from the Cantor set to a Peano space. In our case, we use that we can fit infinitely many Cantor sets into the real line, for example fitting a Cantor set in every interval $[2n,2n+1]$ with integer $n$. Now, as $X$ is $\sigma$-compact, we can define a surjective function $g:C\to X$, where $C$ is the union of infinite Cantor sets.

The second, harder problem comes when we have to extend the continuous function $g:C\to X$ to a continuous function $f:\mathbb{R}\to X$. We have to define the image of $f$ in the countable open intervals that form $\mathbb{R}\setminus C$, in such a way that the function $f$ is continuous. In his proof of 31.5, Willard uses uniform local arc connectedness at this point. What I did at this point was the following:

Given points $a,b\in X$, define $\varrho(a,b)$ to be the infimum of the diameters of the paths from $a$ to $b$. Clearly $\varrho(a,b)>d(a,b)$.

Then, I defined the extension $f$ in each interval $(p_n,q_n)$ to be a path from $g(p_n)$ to $g(q_n)$ with diameter less than $2\varrho(p_n,q_n)$. Defined this way, $f$ will be continuous. However, to prove it is not trivial and requires an epsilon delta argument similar (maybe a bit more complex) than the one in the proof of Willard´s 31.5. This is the hardest part of the proof, in my opinion.

Now the two images with the proof in Spanish: