containment in a cantor set

Question

I said that the following statement was false: If a belongs to the Cantor set and I is an open interval containing a, then I contains at least one other member of the Cantor set.

Unfortunately I am unable to know for sure if I am right or wrong as I do not have the answer key, and am hoping that you could tell me. Thank you.

You are correct: the Cantor set has no isolated points. – Brian M. Scott Mar 19 '13 at 22:01 — Brian M. Scott, Mar 19 '13 at 22:01
No, he's not correct, since he said it was false. – Michael Hardy Mar 19 '13 at 22:04 — Michael Hardy, Mar 19 '13 at 22:04

Dominic Michaelis · Accepted Answer · 2013-03-19T22:10:23.483

2

You can show that every point of the cantor set is an accumulation point with the triadic presentation of the cantor set, so that $$x\in C \iff x=\sum_{k=1}^\infty \frac{a_k}{3^k}$$ with $a_k\in\{0,2\}$. Now you take a sequence converging to this $x$ which is in the cantor set.
By showing that every point is an accumulation point you see that there are no isolated points, so it is right.

edited Mar 19 '13 at 22:10

answered Mar 19 '13 at 22:02

Dominic Michaelis

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containment in a cantor set

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