Compactification and Extension of maps.

[WWGD]

Hi: a couple of questions on (Alexandroff) 1-pt. compactification:

Thanks to everyone for the help, and for putting up with my ASCII posting
until I learn Latex (in the summer, hopefully.)

I wonder if anyone still does any pointset topology. I see many people's
eyes glace when I talk about, e.g., normal spaces. Anyway:

1) I am trying to show that if I have a compact space X and remove an
arbitrary point x_0 from this space, then the resulting space X-x_0 is
locally compact (where I assume that the definition of compact and locally
compact include Hausdorff)


I think we can say that an open subset of a locally compact space is locally compact.
and then we can use the fact that {x_0} is closed, and then apply this.

But I wonder if this is also a way of doing it:
Maybe one can reverse/invert the process of 1-pt. compactification CX of a compact
Hausdorff space X, since CX is Hausdorff iff X is locally compact, i.e., we
start with a compact, Hausdorff space X, (assumed to be the 1-pt. compactification
of some other space Y ) , remove a point x_0 and then Y must have been Hausdorff
and locally compact to start with.


2) Extending functions f :X-->X to functions f^:CX-->CX , with

CX the compactification of X ( i.e., so that f^|_X=f ,with

f^|_X the restriction of f^ to X ) , so that the extension is continuous

or analytic ( I am thinking of CX as the Riemann Sphere, with X =Complex Plane)

Under what conditions can we do this?. All I (think) I know is that

f:X-->X can be extended continuously if f is regular ( inverse image of every

compact set is compact) . What conditions do we need to extend a

function f that is analytic in the complex plane into a function f^ that is

analytic in the sphere, , i.e., when/how can we find f^: Riemann Sphere to

itself, and f^|_X (restriction to complex plane) =f ?.

Thanks.

 

[jvicens]

Thank you for your questions on 1-pt. compactification. Pointset topology is still a relevant and important field in mathematics, and it is great to see someone interested in it.

To answer your first question, I believe your approach is correct. Since X is compact, it is also locally compact. Therefore, an open subset of X is also locally compact. Removing a point x_0 from X will not affect its local compactness, as long as x_0 is not in the open subset. In other words, X-x_0 will still be locally compact.

As for your second question, extending functions from X to CX is not always possible. It depends on the properties of the function f. As you mentioned, if f is regular, then it can be extended continuously. However, for analytic functions, the condition is a bit more complicated. There are several theorems that discuss this, such as the Riemann mapping theorem and the Carathéodory extension theorem. These theorems provide conditions under which an analytic function can be extended to the Riemann sphere. I suggest looking into these theorems for more information.

I hope this helps answer your questions. Good luck with your studies!