Is every region a domain? Am I correct that I understood the definition of domain to be an open, connected set? Does every region have to be a domain? For example: $|z-1+i|\le 3$ is a region if I've understood it correctly, but this is not open and therefore not a domain. Why would this be a region?
closed as not a real question by Michael Lugo, Pete L. Clark, S. Carnahan♦, Anton Geraschenko Jan 18 '10 at 7:27
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Standard definitions in geometric complex analysis are as follows:
A domain is a nonempty open connected set (just as in analysis in general).
A region is a set whose interior is a domain and which is contained in the closure of its interior.
For example the open unit disk and none, part, or all of its boundary (the unit circle).
The closed unit disk together with the interval $[1,2]$ on the real axis is not a region.