I will ask the main question first, and then give the motivation for this one! The question is a bit specific, but seems to be the most general question to ask after handling some obvious counterexamples. I will probably come back and periodically add more questions, hopefully some results, as I look into it.

Q1: Let $X \subset \mathbb{R}^2$ be a one-dimensional Peano continuum. If $X$, aside from its points whose neighborhoods are arcs, has only finitely many local cut points, can $X$ be written as a finite union of dendrites $D_1, \dots, D_n$? If so, does $n$ depend on $X$ or just the number of local cut points?

I assume known what is meant by dendrite and Peano continuum. By dimension, we mean topological dimension, and by a local cut point I mean a point $x \in X$ such that $x$ has an open, connected neighborhood $U$ such that $U \setminus x$ is not connected.

The requirement on the local cut points is necessary by the following counterexample. Let $C_n$ be a circle of radius $\frac{1}{n^2}$ with $n$ chords between its left and rightmost points, and let $X = \cup C_n$ be formed by attaching the rightmost point of $C_n$ and the leftmost of $C_{n+1}$. Then it is quite painless to show that this is a one-dimensional Peano continuum that can not be written as a finite union of dendrites. By attaching two copies of this space together at the left points of $C_1$ and the limit points we get another counterexample with no cut points, just local ones.

I would imagine this has been considered before. I am mostly hoping that this topic can worm its way into a couple open statements, and maybe be a collection point for results of this type. Here are some of the natural extensions:

Q2: Suppose in addition that $X$ is locally cyclicly connected (every point is contained in arbitrarily small neighborhoods $U$ each of whose pair of points is connected by a simple closed curve in $U$). Can $X$ be written as the union of two dendrites? Three?

Q3: (Weaker than Q2) Can the Sierpinski Triangle be written as the union of two (or three) dendrites?

Q4: (Stronger than Q2) If $X$ is locally $n$-connected (every point is contained in arbitrarily small neighborhoods $U$ each of whose pair of points is connected by $n$ arcs in $U$ intersecting only in their end points), can we bound the necessary number of dendrites? Is it $n$ (or $n+1$)?

Q5: What planar Peano continua can be written as the union of two (or three, or finitely many) dendrites each of which intersect at most in a subset of their end points? (We use the order, or Menger, definition of end point; an alternate definition would be to ask the same question with certain restrictions placed on what types of end points we want to allow, e.g. how about just using the free end points, ones with neighborhood an arc?)

Have fun!


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