Examples of homeomorphic topological spaces In my topology course we had some classical examples for homeomorphic topological spaces, e.g. $\mathbb{R}^2/{\sim_{\mathbb{Z}^2}} \cong S^1 \times S^1$ where $x \sim y$ if $x-y \in \mathbb{Z}^2$ and so on. We proved this statement via the quotient topology:
$$\overline{f} \colon X/\sim_f \rightarrow X/ \sim$$
$\overline{f}$ is a homeomorphism if it is continuous, surjective and open on saturated open sets. 
Could you provide other similarly "simple" examples other than $\mathbb{R}^2/{\sim_{\mathbb{Z}^2}} \cong S^1 \times S^1$ and variations thereof?
 A: Here are a couple of important examples that pop up in algebraic topology. For $n\geq 0$, let 
$$S^n := \{(x_1,...,x_{n+1})\in\mathbb{R}^{n+1}\ |\ x_1^2+\cdots+x_{n+1}^2=1\}$$
be the $n$-sphere, and 
$$D^{n+1}:= \{(x_1,...,x_{n+1})\in\mathbb{R}^{n+1}\ |\ x_1^2+\cdots+x_{n+1}^2\leq 1\}.$$
be the $n$-disk. Then, $S^{n} = \partial D^{n+1}\subseteq D^{n+1}$, and the space $D^{n+1}/S^{n}$ is homeomorphic to $S^{n+1}$. The cone of a space $X$ is the quotient
$$CX:= \frac{X\times I}{X\times\{1\}}.$$ 
We have another homeomorphism $CS^n\cong D^{n+1}$. Similarly, the suspension of a space $X$ is defined by:
$$\Sigma{X}:=\frac{X\times I}{X\times\{1\}\cup X\times\{0\}} \cong \frac{CX}{X\times\{0\}}.$$
Then, putting the previous two together it shouldn't be too hard to believe that $\Sigma{S^n}\cong S^{n+1}$.
A: Here is one example of homeomorphism. $$ $$ Let us consider the annulus $ A=\{(x,y) \in \mathbb{R}^{2}|1 \leq x^{2}+y^{2}\leq 4 \}$ and Consider the Cylinder in $ \mathbb{R}^{3} $  given by $$ $$
  $ C=\{(x,y,z)\in \mathbb{R}^{3}x^{2}+y^{2}=1, 0\leq z \leq 1 \}$ .
Then there exists  continuous functions $$ $$ $ f:A \rightarrow C \ and \ g:C \rightarrow A $  defined by $$ $$ $ f(x,y)= \left(\frac{x}{\sqrt {x^{2}+y^{2}}},\frac{y}{\sqrt {x^{2}+y^{2}}},\sqrt {x^{2}+y^{2}}-1 \right) $  and $ g(x,y,z)=\left(  (1+z)x,(1+z)y \right) $ . Little more calculation shows $ f \circ g=g\circ f=I $
. Hence f and g are homeomorphism.
