# Is there a mathematical notation for the region enclosed by a set of curves

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. • 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? – CiaPan Nov 23 '20 at 16:45

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.)