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How many components of complement of a compact and connected set in $\mathbb{R^2}$has?

Are there any tricks to solve the problem?

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The complement of a compact, connected set in $\Bbb R^2$ can have any finite number of components, and it can also have countably infinitely many components. The complement of a line segment has one component. The complement of a circle has two. The complement of a figure $8$ has three. By adjoining more and more circles, you can get any finite number. Finally, if you take the union of circles of radius $\frac1n$ and centre $\left\langle\frac1n,0\right\rangle$, you get a compact, connected set whose complement has infinitely components.

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It could be anything. Consider a closed disc with $n$ separated open discs deleted from it. Such a set is closed and bounded, hence compact; it is clearly connected. But its complement has $n+1$ connected components.

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