Why is $D^{n+1}/S^{n} = S^{n+1}$ true? I went to my first lecture in Algebraic Topology and managed to get really confused. It seems like they assumed that the following statement was "obvious":
$D^{n+1}/S^{n} = S^{n+1}$
Where $D^{n}$ is the unit disk/ball in $\mathbb{R}^{n}$ and $S^{n}$ is the unit sphere in $\mathbb{R}^{n+1}$.
The notation $X/A$ is, as far as I could understand, a way of denoting that $A\subseteq X$ collapse to one point. So we divide $X$ into equivalence classes where the points in $A$ are in the same equivalence class and the points $X-A$ is in their "own" equivalence class.
My questions:


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*Why is this statement true? and do $=$ indicate that there exists a homeomorphism between the two spaces?


Here's one way to set up a concrete homeomorphism.
The sphere $S^{n+1}$ is homeomorphic to the $1$-point compactification of $\mathbb{R}^{n+1}$. This is witnessed by the stereographic projection maps. Take the disc $D^{n+1} \subset \mathbb{R}^{n+1}$, map the inner open disc to $\mathbb{R}^{n+1}$ (you have probably seen this done in a point-set topology class), and map the boundary of $D^{n+1}$ (which is $S^{n}$ by definition) to the point at infinity. This is not injective, but if you define the same map on $D^{n+1} / S^{n}$, this map is bijective and continuous (not too hard to see, recall which sets are open in the compactified $\mathbb{R}^{n+1}$), and goes from a compact space to a Hausdorff space, hence is a homeomorphism (another standard point-set result).
Another idea:
It is relatively easy to show that the sphere is two discs glued at their boundaries.
Let $X_1$ be an inner disc of half the radius of $D^{n+1}$ inside $D^{n+1}/S^{n}$. Let $X_2 = D^{n+1}/S^{n} - X_{1}$. We know that $X_1$ is a disc, and $X_2$ looks like an annular region with collapsed boundary. Define a map from $X_2$ to a disc in the following way. Map rings on the inside of $X_2$ to rings on the outside of a disc, where by rings I mean copies of $S^n$ defined as level sets of the Euclidean norm.
Do this inside-out, so that rings on the outside of $X_2$ get mapped to smaller rings in the interior of a disc. Finally, send the outer, collapsed boundary of $X_2$ to the center of the disc. This is clearly a bijection. It is not too hard to show it is continuous. It is a map from a compact space to a Hausdorff space, hence a homeomorphism, so $X_2$ is actually a disc.
Thus, $D^{n+1}/S^n$ was just two discs glued together at their boundaries, hence is homeomorphic to $S^{n+1}$.
