Homeomorphisms

Definition (Homeomorphism)

Let $f:X\to Y$ be continuous and bijective. We say $f$ is a homeomorphism if $f^{-1}:Y\to X$ is continuous.

Lemma (Homeomorphism Retains Continuity Structure)

Let $f:X\to Y$ be a continuous bijection. Then $f$ is a homeomorphism

$$\begin{array}{rl}& ⟺\forall \text{ open }O\subset X,f(O)\text{ is open in }Y\\ & ⟺\forall \text{ closed}F\subset X,f(F)\text{ is closed in }Y\end{array}$$

Proof: Since $f$ is a bijection, $g:=f^{-1}:Y\to X$ is well defined. By lemma, we have that $g$ is continuous iff $g^{-1}(O)$ is open for all $O$ open, and vice versa for closed.

Note $g^{-1}=(f^{-1}{)}^{-1}=f$. So the above statement holds for $f$.

Proposition (Minimum for Homeomorphism)

Let $X$ be a compact metric space. Suppose $f:X\to Y$ is injective and continuous. Then $f^{-1}:f(X)\to X$ is continuous. There is the same as saying $X$ is a homeomorphism from $X$ to $f(X)$.

Proof: Note that $f:X\to f(X)$ is continuous and bijective (since it must surjective to its image). To see that $f^{-1}$ is continuous, we need to show the preimage of closed sets are closed by proposition. Let $F\subset X$ be closed and consider

$$(f^{-1}{)}^{-1}(F)=f(F)$$

Note $F$ is closed, $F\subset X$ and $X$ is compact. So $F$ must be also compact by lemma. Then $f(F)$ is compact by proposition and and by Heine-Borel Theorem, $f(F)$ is closed.

Corollary

Any continuous bijection between compact metric spaces is a homeomorphism.

Theorem (No Dimensional Reduction)

There does not exist a continuous injective map from $[0,1]\times [0,1]$ into $\mathbb{R}$.

Proof: Suppose there is such an $f:[0,1]\times [0,1]\to \mathbb{R}$ that is injective and continuous. Let $I=f([0,1]\times [0,1])$.

1. Since $[0,1]\times [0,1]$ is compact, $I$ is compact
2. Since $[0,1]\times [0,1]$ is connected, $I$ is connected. Therefore $I=[a,b]$ for some $a,b\in \mathbb{R}$. Since $f$ is injective, then $a<b$. Let $c\in (a,b)$ and $f^{-1}(c)=p\in [0,1]\times [0,1]$. But then

$$f:\left([0,1]\times [0,1]\right)⟶[a,c)\cup (c,b]$$

which is a continuous function from a connected set to a disconnected set, a contradiction.