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Suppose we have a metric space $X$ and a continuous injection from $[0,1)$ onto $X$. The case I had in mind will satisfy that $X$ is compact, but the problem I have can be stated more generally, as well.

Examples of such a space include a circle, a circle with a free arc sticking off it, and the Warsaw Circle, which is obtained from the topologist's sin curve by adding a path from the bottom of the limit arc to the starting point of the 'squiggle'.

What I am wondering is what the preimages of arcs in $X$ are allowed to look like in $[0,1)$. Without the injectivity assumption it would be highly permissive, but there are theorems concerning spacefilling curves that say a lot of pathological behavior is impossible with injective maps.

Especially, I have three questions in mind. They feel true to me, but it is not at all clear how to prove the first one. The other two can be worked from the first, as far as I can tell.

1) Given an arc in $X$, must its preimage just be the union of one or two arcs (picture the circle), possibly degenerate? Here I am letting 'arc' mean any connected subset of the line.

2) If $X$ contains a simple triod, i.e. three closed arcs joined into a 3-pointed asterisk by identifying an end point from each, then is $X$ just the circle with a free arc sticking off it?

3) $X$ does not contain a simple $4$-od (defined as expected).

Basically, are 'transverse' arcs possible?

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  • $\begingroup$ 1) is true. It is trivial since the injection $f$ is actually a homeomorphism onto its image for any compact interval, so sends connected sets to connected sets. If there were at least three connected sets in the preimage of some interval, then two would be contained in some such interval. $\endgroup$ – John Samples Jun 21 '18 at 11:48
  • $\begingroup$ 2) is true but non-trivial. mathoverflow.net/questions/302993/… $\endgroup$ – John Samples Jun 21 '18 at 11:48
  • $\begingroup$ 3) Is true by the above classification theorem in the link. $\endgroup$ – John Samples Jun 21 '18 at 11:52

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