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Partitions of the Unit Circle

I have been asked the following question:

Construct a countable infinite partition of the unit circle \mathrm{Construct\ a\ countable\ infinite\ partition\ of\ the\ unit\ circle}
Is it possible to construct such a partition of sets of equal length? \mathrm{Is\ it\ possible\ to\ construct\ such\ a\ partition\ of\ sets\ of\ equal\ length?}

I know that 2π=n=1l(An)=l(An) 2\pi = \displaystyle\sum_{n=1}^{\infty} l(A_n) = \infty * l(A_n)

So do I need to have an infinite set that gives me 2?

I know that 1+1/2+1/4+1/8+1/16+...=2 1 + 1/2 + 1/4 + 1/8 + 1/16 + ... = 2

So could I have a partition of the unit circle given by:

A1A2A3...An... A_1 \cup A_2 \cup A_3 \cup ... \cup A_n \cup ...

Where {A1,A2,A3,...,An,...} \left\{A_1, A_2, A_3, ... , A_n, ...\right\} ={π, π/2, π/4, ...} = \left\{\pi,\ \pi/2,\ \pi/4,\ ... \right\}

Would this work?
(edited 10 years ago)
Hi there,

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