I think the statement that a domain (open connected set) in a sphere is simply connected if and only if its complement is connected is a standard result. But how can one prove it? Is it possible to prove this without algebraic topology?
Asked
Active
Viewed 1,498 times
5
-
Note that by puncturing the sphere, you can essentially consider the same problem for (bounded) open sets in the plane. – Dan Rust Jan 17 '14 at 01:42
-
Only if? if you cut out two disks the remaining piece is connected... – user68061 Jan 17 '14 at 15:32
-
1@user68061 The remaining piece is not simply connected. – Spook Jan 20 '14 at 11:26
-
1Maybe your question is a particular case of this – Alex Ravsky Jan 21 '14 at 19:19