Does There Exist a Planar, Linear, Triodic Tree-Like Continuum?

Motivated by Continuum image of line is chainable?

A planar continuum $X$ is a compact, connected subset of the plane. It is linear if there is a continuous bijection from $[0,1)$ onto $X$, for example a Warsaw circle. It is triodic if it contains a simple triod, i.e. three closed arcs all attached at an end point (3-pointed asterisk). It is tree-like if there is a sequence of surjections $f_n : X \rightarrow T_n$ onto finite trees such that max$[\text{diam}(f_n^{-1}(x))]$ goes to zero.

I strongly suspect no such object exists. It seems like the 'being tree-like' happens after finite time and then the tail will create a loop in the plane when it comes back around to start converging back onto itself. The inclusion of a triod then seems like it will cause impossibilities for the bijectiveness criterion of linearity.

Any thoughts? I found very little overlap in the investigation of linear continua and treelike continua.

• It seems that each linear continuum containing a triod is homeomorphic to the circle with an attached arc, so cannot be tree-like. In the proof one should use the local connectedness of the triod in its central point. – Taras Banakh Jun 17 '18 at 16:53
• Do you have a reference? The dimensionality issue seems to require some subtle facts, e.g. right off the bat we see that such a statement implies things about injectivity and spacefilling curves. – John Samples Jun 18 '18 at 1:03
• For a linear continuum the set of accumulation points of the half-line is a compact connected subset $A$ of the half-line. Depending on the form and position of this connected subset, we can distinguish 4 possible topological types of linear continua: if $A$ is a singleton, then the linear continuum is either the circle or the circle with an attached arc; if $A$ is a non-degenerated interval, then the linear continuum is either Warsaw circle or the Warsaw circle with an attached arc at its end-point. It would be nice to prove such a classification theorem. – Taras Banakh Jun 18 '18 at 15:00
• @TarasBanakh Actually, your statement about circles/Warsaw circles is Lemma 4 in "Continua which are a one-to-one continuous image of $[0,\infty)$" by Sam Nadler. – D.S. Lipham Jun 19 '18 at 1:45
• @David Great! Then it remains to prove that there exists 5 pairwise inequivalent embedding of linear continua in the plane: the Warsaw circle has two non-equivalent embeddings. – Taras Banakh Jun 19 '18 at 4:41