1. ## 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.

2. ## 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.

3. ## 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)?