Hatcher - simplicial and quotient representations of spheres I am reading chapter 2 of Hatcher's algebraic topology text. In it, he makes the following two claims:
1) Quotienting $D^n$ by $\partial D^n$ produces a space homeomorphic to $S^n$.
2) We may create a simplicial complex homeomorphic to $S^n$ by taking two copies of $\triangle^n$ and identifying each corresponding pair of vertices to a single point (and hence identifying the corresponding edges).
These both seem intuitively clear to me in low dimensions. For example, for the second part, taking two line segments and gluing them together at their ends produces a circle. However, I do not see how to prove these statements rigorously. I tried to find explicit homeomorphisms to $S^n$, but failed. 
These seems like basic facts, but I could not find justifications in chapter 0 or what I've read of chapter 2. I would appreciate bulletproof, rigorous proofs, or indications of how to write them. The half-baked arguments I'm coming up with are quite handwavey. 
 A: For rigorous proofs, you will need to remember your point-set topology:
1) Consider the map $\phi:D^n \rightarrow S^n$ that "wraps" the disk around the sphere. If you use the standard definitions: $D^n = \{\mathbf{x} \in \mathbb{R}^{n}\mid \|\mathbf{x}\| \leq 1\}$, and $S^n = \{\mathbf{x} \in \mathbb{R}^{n+1}\mid \|\mathbf{x}\| = 1\}$, you can construct $\phi$ explicitly:
$$\phi(\mathbf{x}) = (2(1 - \|\mathbf{x}\|^2)^{1/2}\mathbf{x},1 - 2\|\mathbf{x}\|^2)$$
Now, it is clear that $\phi$ is continuous, and it is simple to check that $\phi$ does actually map $D^n$ surjectively onto $S^n$, and it is injective except it sends the entire boundary to one point on the sphere. So, the last thing you need to know is that $\phi$ is a quotient map. But this is easy, since all closed maps are quotient maps, and all continous maps from a compact space to a Hausdorff space are closed. Then $D^n$ is compact, $S^n$ is Hausdorff, $\phi$ is continuous, and you are done.
2) You need to construct two maps from the standard complex $\Delta^n$ into the sphere $S^n$. First, get a homeomorphism $\phi:\Delta^n \rightarrow D^n$ by "rounding out the edges". Now, there are a couple of ways to do this concretely, I'll let you figure one out. Once you have that, you get two embeddings from $D^n$ to the sphere: mapping to the upper half $i_U$ and the lower half $i_L$:
$$i_U(\mathbf{x}) = (\mathbf{x},\sqrt{1 - \|\mathbf{x}\|^2})$$
$$i_L(\mathbf{x}) = (\mathbf{x},-\sqrt{1 - \|\mathbf{x}\|^2})$$
Now this gives you a map:
$$\Delta^n \sqcup \Delta^n \rightarrow D^n \sqcup D^n \rightarrow S^n$$
Where we use $\phi$ the first time, and $i_U,i_L$ on the two components in the second map. The key here is that both $i_U$ and $i_L$ agree on the boundary of $D^n$, and these will correspond (under $\phi$) to the points/edges/etc. of $\Delta^n$ that we were trying to identify!
Tell me if you'd like more clarification.
