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Let S be a set of pairwise disjoint 8-like symbols on the plane. (The 8s may be inside each other as well) Prove that S is at most countable.

Now I know you can "map" a set of disjoint intervals in R to a countable set (e.g. Q :rational numbers) and solve similar problems like this, but the fact that the 8s can go inside each other is hindering my progress with my conventional approach...

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    I stand corrected.2014-09-29

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Let $\mathcal{E}$ denote the set of all your figure eights. Then, define a map $f:\mathcal{E}\to\mathbb{Q}^2\times\mathbb{Q}^2$ by taking $E\in\mathcal{E}$ to a chosen pair of rational ordered pairs, one sitting inside each loop. Show that if two such figure eights were to have the same chosen ordered pair, they must interesect, which is impossible. Thus, $f$ is an injection and so $\mathcal{E}$ is countable.

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    If figure eight $A$ is inside a loop of figure eight $B$, $A$'s index points cannot include a point from the other loop of $B$. If $A$ and $B$ are external to each other, then they can have no index points in common.2011-11-03
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Hint: Suppose we have a lower bound on the size of the 8s. Show that no two of them can be very close, and conclude that there are only countably many of them.

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    It does work for other figures :)2011-11-02