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Math Help - Topology Question

  1. #1
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    Topology Question

    I'm reading Hatcher's algebraic topology book... and this is probably a stupid question... but I can follow the basic definitions, but in the exercises he refers to spaces like D^2 and S' and I don't see a place where he defines what properties these spaces have. Do you know a website that defines these basic spaces so that I'll have a bit of background knowledge (I searched D2 disk and the whatnot on google, but I didn't find anything useful). ...I guess this is something I should have learned in an undergraduate course, but it never came up.
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  2. #2
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    Re: Topology Question

    usually D^2 is just a 2-dimensional disk. the most general way to represent it in the Euclidean plane is

    D^2 = \{(x,y) \in \mathbb{R}^2: (x-h)^2 + (y-k)^2 \leq r^2\}, where the inequality is strict if it is an open disk.

    for convenience the center (h,k) is often taken to be (0,0) and the radius taken to be 1.

    i am guessing that the other set is S^1 = \{(x,y) \in \mathbb{R}^2: x^2 + y^2  = 1\} or its equivalent complex form,

    S^1 = \{z \in \mathbb{C}: |z| = 1\}, the unit circle, which is the boundary of the unit disk.
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  3. #3
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    Re: Topology Question

    Ohh, okay, thanks. And would R^3 just be a 3-dimensional plane? What would D2 U D2 be? (I know S1 U S1 is the torus... but I'm not sure about D2 U D2)?
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