Can we find a homeomorphism from the square $Q_2$ of side length $2$ centered on the origin and the unit circle $S^1$?
We can easily define a map $r:Q \longrightarrow S^1$ by
$$(x,y) \mapsto \bigg(\frac{x}{\sqrt{x^2+y^2}},\frac{y}{\sqrt{x^2+y^2}} \bigg)$$ which is a radial projection onto the unit circle, but how can we define $r^{-1}$ and show that it is continuous?
Thoughts
I think we may define the inverse map as
$$(x,y) \mapsto \bigg(\frac{x}{\sqrt2\max{│x│, │y│}} , \frac{y}{\sqrt2\max {│x│, │y│}}\bigg)$$At least intuitively this maps to a square, and the $\frac{1}{\sqrt2}$ term scales appropriately. However, how can we demonstrate these are continuous maps, thus demonstrating that $Q_2$ and $S^1$ are homeomorphic?
Any help would be appreciated. Regards, MM.
Best Answer
Let me outline a proof; I shall leave the details of this proof as exercises.
Exercise 1: Prove that the unit circle $S^1$ is homeomorphic to the space obtained from the unit interval $[0,1]$ by identifying the endpoints $0$ and $1$. (Hint: the exponential mapping $\theta\to e^{i\theta}$ can be composed with a linear mapping to give a homeomorphism.)
Let $C$ be the image of a simple closed curve in the plane, that is, let $C$ be the image of a continuous function $f:[0,1]\to \mathbb{R}^2$ such that $f:(0,1)\to\mathbb{R}^2$ is injective and $f(0)=f(1)$.
Exercise 2: Prove that $C$ is homeomorphic to the unit circle $S^1$. (Hint: show that $f:[0,1]\to C$ induces a continuous bijection $\tilde{f}:S^1\to C$ using Exercise 1 and then show that $f$ is an open mapping using the compactness of $S^1$ and the fact that the plane is Hausdorff.)
We now need an elementary lemma of point-set topology:
Exercise 3: Let $X$ be a topological space and let $X=A\cup B$ where $A$ and $B$ are closed subspaces of $X$. Prove that if $g:A\to Y$ and $h:B\to Y$ are continuous functions into a topological space $Y$ such that $g(x)=h(x)$ for all $x\in A\cap B$, then there is a unique continuous function $f:X\to Y$ such that $f(a)=g(a)$ and $f(b)=h(b)$ for all $a\in A$ and $b\in B$.
Finally, we can prove the result of your question:
Exercise 4: Prove that the unit square is the image of a simple closed curve in the plane and conclude that it is homeomorphic to the unit circle. (Hint: you can use Exercise 3 to "glue" continuous mappings together.)
Exercise 5: Give examples of images of simple closed curves in the plane (using Exercise 3) and conclude (using Exercise 2) that they are homeomorphic to the unit circle $S^1$.
I hope this helps!