
#1
Mar1709, 07:54 AM

P: 391

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


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