An open connected set is called a domain " " : Carath�odory considers obviously disjoint sets.
2.
Such a path exists since " U " is assumed to be an open connected set.
3.
According to Hans Hahn, the concept of a domain as an open connected set was introduced by Constantin Carath�odory in his famous book.
4.
Here a major difference is evident from the one-variable theory : while for any open connected set in we can find a function that will nowhere continue analytically over the boundary, that cannot be said for.
5.
For example, in his influential monographs on elliptic partial differential equations, Carlo Miranda uses the term " region " to identify an open connected set, and reserves the term " domain " to identify an internally connected, perfect set, each point of which is an accumulation point of interior points, according to this convention, if a set is a region then its closure " } } is a domain.
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