- #1
AxiomOfChoice
- 533
- 1
Definition of "compactness" in the EXTENDED complex plane?
How does one define a compact set in the extended complex plane [itex]\mathbb C^* = \mathbb C \cup \{ \infty \}[/itex]? "Closed and bounded" doesn't really make sense anymore, as I'm assuming it's permissible for a compact set to contain the point at infinity...right? I guess the "finite subcover" definition still holds, as always, but this doesn't seem very useful. Are there other, more helpful, equivalent characterizations for compact subsets of [itex]\mathbb C^*[/itex]?
How does one define a compact set in the extended complex plane [itex]\mathbb C^* = \mathbb C \cup \{ \infty \}[/itex]? "Closed and bounded" doesn't really make sense anymore, as I'm assuming it's permissible for a compact set to contain the point at infinity...right? I guess the "finite subcover" definition still holds, as always, but this doesn't seem very useful. Are there other, more helpful, equivalent characterizations for compact subsets of [itex]\mathbb C^*[/itex]?