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Compactification and Extension of maps.

  1. Mar 17, 2009 #1

    WWGD

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    Gold Member

    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.
     
  2. jcsd
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