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Can the Sierpinski Triangle be written as a finite union of dendrites? If so, can it also be verified what the minimal number is (assuming you can't do it with just two)?

This is a small piece from a more sophisticated thread, but the particular example can possibly be solved by 'geometric cleverness' so I wanted to make it a separate question. The main thread is:

Finite Unions of Dendrites

John Samples
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  • Apparently there is a connected subset of the Sierpinski Triangle which is not path-connected. So to prove that it can't be written as the finite union of dendrites, is this true: If $D$ is a finite union of dendrites, is every connected subset of $D$ path-connected? Note that a dendrite itself satisfies this, so an induction argument would work.

    Aside from this, after finally playing around with this problem for a bit I was already getting the impression that it's impossible.

     – John Samples Dec 19 '20 at 14:13

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No, it can't be written as a finite union of dendrites. Actually, it can't be written as a countable union of dendrites. Since every neighborhood in the Sierpinski Triangle $S$ contains a simple closed curve, a dendrite has empty interior in $S$, so by the Baire Theorem you can't cover $S$ with countably many dendrites. This is also true of dendroids, $\lambda$-dendroids etc. by the same argument; any class of continua which are acyclic.

This Baire trick was too clever for me, a friend hit me up with this one.

John Samples
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I have probably horribly misunderstood the question but here goes:

if you consider a triangle fractal (https://i.sstatic.net/shOK1.gif) with depth 1 then there are 3 dendrites, depth two there are 6 dendrites, depth three there are 15 dendrites.

so the Sierpinski triangle is this triangle fractal with depth n=inf. The formula for the number of dendrites for a triangle fractal with depth n is: $$ 3+3^1+3^2+3^3+...+3^{n-1}=-3(1-3^n)/2 $$

the limit as n->inf for the nth term diverges to infinity, so the Sierpinski triangle can't be written from a finite amount of dendrites.

Lmnop
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  • Avoid adding links to images. You can directly add the image in the question body. – Anindya Prithvi Dec 19 '20 at 17:23
  • Hey! Is this a formal concept, the "dendrite fractal depth" or such? – John Samples Dec 20 '20 at 11:56
  • I don't think so. But I think you could formalize the argument. Am I understanding what the question is asking or am I on a completely wrong track? I haven't done topology but I thought this Q looked interesting enough. – Lmnop Dec 20 '20 at 11:58
  • By depth, I meant layers of self-similarity which I think is a formal concept but again I'm not sure. – Lmnop Dec 20 '20 at 12:00
  • I spent a bit of time trying to build the dendrites manually; yeah, the space is just too 'deep' to get everywhere without having to make some loops. – John Samples Dec 20 '20 at 12:04