# Spaces That Have Uncountably Many Disjoint Copies in $\mathbb{R}^2$

There is a theorem by Moore that says there are not uncountably many disjoint copies of the simple triod in the plane (the simple triod is the space by adjoining one end point from three copies of $[0,1]$ together to make a three-pointed asterisk). I am wondering what other compact, connected spaces also satisfy this.

More specifically, for any compact subset of the plane there always exist countably many disjoint copies (easy to see taking disjoint discs and the fact that they are homeomorphic to the plane), but which ones don't have uncountable such collections? Of course any space must be one-dimensional, except the example of a singleton, and none can contain a simple triod.

For locally connected, compact, connected subsets of the plane, if they are not an arc or a circle then they contain a simple triod, by some standard theorems (they are locally path-connected), so examples need to be a bit gross. Just by intuition it seems like the sin$(\frac{1}{x})$ continuum (closed topologist's sine curve) does not work, maybe because it has some similarity to a triod.

There are some results using the concepts of span, but I am wondering if there are any other sort of . . . sporadic examples that are easy to see naturally.

• What a cool theorem by Moore. – Cheerful Parsnip Jul 6 '18 at 5:39
• To start with, the "triods" for which Moore proved his theorem are more general than the "simple triods" you mention in your question. – bof Jul 6 '18 at 9:44
• Bof is right, I had only ever read the Pittman paper, a simplified version of the proof. Never realized the original was stronger. I found a nice 'colored' version in the Pommerenke book 'Boundary Behavior of Conformal Maps', though it doesn't give another example, just expands the inadmissible class of triods from totally disjoint to 'partially disjoint with conditions'. The original Moore Theorem covers the case where the arcs are merely assumed to be nondegenerate irreducible continua. Not sure if the colored version holds in this case, would have to check the paper. Easy publication? – John Samples Jul 6 '18 at 12:38