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Math Help - Continuous surjections

  1. #1
    Senior Member bkarpuz's Avatar
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    Continuous surjections

    Dear MHF members,

    I have the following problem.
    Let \mathbb{D}:=\{(x,y):\ x^{2}+y^{2}\leq1\} denote the unit disk in the plane, with interior \mathbb{D}^{\circ} and boundary \partial\mathbb{D}.

    1. Does there exist a continuous surjection f:\mathbb{D}\to\mathbb{D}^{\circ}?
    2. Does there exist a continuous surjection f:\mathbb{D}^{\circ}\to\mathbb{D}?
    3. Does there exist a continuous surjection h:\mathbb{D}\to\partial\mathbb{D}, which leaves every point on \partial\mathbb{D} fixed?

    Many thanks.
    bkarpuz
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  2. #2
    Senior Member Tinyboss's Avatar
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    1) The continuous image of a compact set is compact.
    2) Yes. Think about expanding the disk a little, and "capping" the radius of the image at 1.
    3) No, but I don't know of a really short proof.
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