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I want to define the region $S$ as the area enclosed by the curves $C_1, C_2, C_3, C_4$ as in the picture below, but do not know what notation to use. Any help would be appreciated.

enter image description here

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  • $\begingroup$ Suppose these three colourful things are curves (circles) in a plane, and you devise a name – which one of the three shapes would it apply to? $\endgroup$ – CiaPan Nov 23 '20 at 16:45
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AFAIK there is no standard notation for this. Best to say it in words.

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  • $\begingroup$ I see. Thank you for the help. $\endgroup$ – Salo Nov 23 '20 at 16:30
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The blue path is the boundary of $S$, denoted $\partial S$. We cannot define an inverse operator $\partial^{-1}$ to apply to this boundary so as to recover $S$, because $S$'s complement $S^\complement$ has the same boundary, i.e. $\partial S=\partial S^\complement$. However, you can disambiguate which set with the given boundary is intended. An important generalization of your example is the interior of a Jordan curve, which has been discussed here before. In one notation, $S=\operatorname{int}(\partial S)$ (see e.g. p.4 here). (As to whether that notation serves your needs, we should address an ambiguity in your question: $\operatorname{int}(\partial S)$ doesn't include $\partial S$ itself, but $\overline{\operatorname{int}}(\partial S):=\partial S\cup \operatorname{int}(\partial S)$ does.)

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  • $\begingroup$ Thanks, it's very helpful. $\endgroup$ – Salo Nov 23 '20 at 16:54

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