Let $\theta$-set be the union of a circle with one of its chords. Show that any collection of disjoint $\theta$-sets in the plane is countable?

I cannot understand the question and I do not know where to start with, could anyone kindly help? Thanks.


A chord is just a straight line segment between two points on the circle (like in the letter $\theta$!)

There is a standard trick to answer these problems but I actually think it's instructive to consider first just a disjoint collection of circles in the plane. Must this collection be countable? The answer is 'no'. If you understand the example of an uncountable such family, you will begin to see why the addition of chords to each of the circles must help.

  • $\begingroup$ Could you kindly tell the difference between the two situations (circle vs $\theta$)? I have thought on it but quite do not see the true reason. That has to be related to the fact that the chord is "straight" but I fail to see precisely the details. And what happens to "swallows" (i.e. degenerate Y's, like $\curlyvee$)? Thank so much for your help. $\endgroup$ – Romeo Apr 10 at 7:18

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