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The Denjoy-Riesz Theorem states that any compact zero-dimensional subset of the plane can be covered by an arc, i.e. an embedded image of $[0,1]$. Sometimes it's stated just for covering a Cantor Set, and there's also a generalization by Moore and Kline:

If $M \subset \mathbb{R}^2$ is compact, then it's contained in an embedded arc if and only if each component of $M$ is a point or a simple arc $A$ such that if $x \in A$ is a limit point of $M \setminus A$ then $x$ is an endpoint of $A$.

Does anyone know a modern reference for the Denjoy-Riesz Theorem?

Either a book, or a paper giving some modernized proof. I don't mind if it's just the Cantor Set version. It's not in Arkhangel'skii or Novikov. The only two sources I've ever seen this proved in are Kuratowski's tome, where it's derived from a mountain of other theory that's only tangentially related, or in Moore's paper which is nearly unreadable. It references many technical results from a previous paper, and together it'd probably be 20 pages of math just for this one theorem.

Has anyone ever encountered this theorem in a book with a proof?

EDIT: I'm also trying to just prove it myself in the meantime. I can prove it in the following case for a Cantor Set $C$:

Suppose we are given a nested sequence of finite partitions $\mathcal{P}_i$ of $C$ into clopen sets (so, each element of $\mathcal{P}_i$ is partitioned into finitely many sets in $\mathcal{P}_{i+1}$) and the diameters of the elements of $P_i$ converge uniformly to zero as $i \rightarrow \infty$. Then $C$ is covered by an embedded arc.

The proof is very similar to the proof of the Hahn-Mazurkiewicz Theorem using property $S$ and the hyperspace of closed sets; it relies only on the Schoenflies Theorem and the Zoretti Theorem (that for any two compact $A, B$ in the plane there's a Jordan curve separating them). Just start chopping the space up with nested Jordan curves, drawing arcs between the boundaries at each step, and approximating uniformly. The end result might not a priori be an embedding, but you can appeal to the fact that a path-connected Hausdorff space is arc-connected, take an arc going from one 'end point' to the other, and show that it contains all of $C$ immediately from the construction.

That's my idea for a proof, please feel free to point out problems that might arise, or how to make further progress!

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  • $\begingroup$ In fact the Schoenflies is only used in a weak way, namely that every point on a Jordan curve is arcwise-accessible. $\endgroup$ – John Samples Feb 21 at 21:52
  • $\begingroup$ Basically, one should have that every Cantor Set in the plane has such a sequence of partitions - but I'm having trouble showing it. $\endgroup$ – John Samples Feb 21 at 22:06
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I ended up just proving it myself here: https://math.stackexchange.com/questions/4027493/accessible-proof-of-denjoy-riesz-theorem

How to extend these methods to the Moore-Kline Theorem is still unclear, please take a look if you think that you might be able to help!

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