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Math Help - Algebraic Topology - Motivation for homeomorhism between disk and R2

  1. #1
    Super Member Bernhard's Avatar
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    Algebraic Topology - Motivation for homeomorhism between disk and R2

    In his book Essentail Topology, Martin Crossley deals with an example involving the closed disk D (see attachment)

     D^2 = \{ (x,y) \in \mathbb{R}^2 : \surd (x^2 + y^2)  \leq 1 \}

    Crossley notes on page 78 (see attachment) that  D^2 - \partial D^2 is homeomorphic with  \mathbb{R}^2 , by the mapping


     (r cos \ \theta, r sin \ \theta ) \longrightarrow ( tan  \ \frac{\pi r}{2} \ cos \ \theta \ , \ tan \ \frac{\pi r}{2} \ sin  \ \theta )

    However, Crossley just states this homeomorphism and gives no sense by which it was derived or discovered. Can someone help me get a sense of why, intuitively speaking, one would come up with such a mapping - why it is intuitively reasonable! i.e. some motivation as to why this mapping would be a sensible or appropriate one to try.

    I am working on proving that it is a homeomorphism - but this is not the same as seeing why it would be appropriate or understanding how and why someone came up with this mapping.

    Can someone please help?

    Peter
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    Re: Algebraic Topology - Motivation for homeomorhism between disk and R2

    conceptually, you can see that the "angle part" remains the same, all we are doing is "stretching" or "shrinking" the radius.

    this is why the tangent function is used. what we are doing is mapping [0,1) to [0,∞). note that between 0 and π/2 tangent takes on all positive values. so all we need to do to map [0,1) to [0,∞) is to find some function that maps [0,1) to [0,π/2). the function f(x) = (π/2)x, will do nicely, so for a function that maps [0,1) to [0,∞) we simply take:

    tan○f
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  3. #3
    Super Member Bernhard's Avatar
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    Re: Algebraic Topology - Motivation for homeomorhism between disk and R2

    Thanks Deveno

    Most helpful!

    Peter
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