- #1

Bacle2

Science Advisor

- 1,089

- 10

**Extending complex functions f:C-->C into f^:C^-->C^**

Hi, All:

I have a function f: ℂ-->ℂ' , i.e., a complex-valued function (use ℂ,ℂ' to make a distinction between the complexes as domain and codomain respectively), and i want to extend it into

the Riemann Sphere ℂ^, i.e., I am looking for f^ such that f^|

_{ℂ}=f. If I remember correctly, a necessary and sufficient condition for extensibility is

that f must be a proper function, i.e., that for every K compact in ℂ' is sent to a compact

set in ℂ, i.e., f

^{-1}(K) is compact in ℂ? I think this somehow had to see with

ℂ^ being the 1-pt compactification of ℂ; is this correct? If so, anyone have a ref. for the

proof, if not, could someone please let me know what the correct result is?

Thanks.