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Math Help - The open long ray

  1. #1
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    The open long ray

    Hello,

    The open long ray L^{+}=w_{1}\times[0,1)-{(0,0)}. My question is how can we show that the open long ray is Hausdorff locally homeomorphic to the real line R, so that it is a manifold of 1-dimension. Also, how can we show that it is normal but not metrizable.

    Please guide me

    Thank you in advance
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by student2011 View Post
    Hello,

    The open long ray L^{+}=w_{1}\times[0,1)-{(0,0)}. My question is how can we show that the open long ray is Hausdorff locally homeomorphic to the real line R, so that it is a manifold of 1-dimension. Also, how can we show that it is normal but not metrizable.


    Thank you in advance
    What have you tried so far--this is just a get down and dirty problem if you know what I mean?
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  3. #3
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    What I tried is the following:

    1. The open long ray is normal since all ordered spaces are completely normal.

    2. The open long ray is not metrizable, since it is not Lindelof ( the collection {(0,\alpha) \alpha \in \omega _{1}} is an open cover of L^{+} having no countable subcover.

    But I miss two points, I should also verify that the open long ray is connected and locally homeomorphic to R, in order to say that it is a manifold. How can I show that.

    Thank you very much
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  4. #4
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    The open long ray

    Hello;

    I am now studying the open long ray, and I proved some properties, for example the open long ray is path connected, hence connected. Kindly, I want from you to check if what I wrote is logic or not.

    Please find attached to my work.

    Thank you in advance
    Attached Files Attached Files
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