I have been asked to proof the following as an exercise like this one:
Let $K$ be the Cantor set and $\tilde{K}$ compact without interior points and no isolated points. Show that there is an increasing homeomorphism $f: \mathbb{R} \to \mathbb{R}$ such that $f(K) = \tilde{K}$.
My attempt and idea to proof it:
The Cantor set can be written as $[0,1] - \displaystyle\bigcup_{n \in \mathbb{N}} I_{n} $, where $I_n$ is the half third of each interval on the previous step. The set $\tilde{K}$ has a $\inf \tilde{K} = m $ and $\sup \tilde{K} = M$ and can be written as $[m,M] - \displaystyle\bigcup_{n \in \mathbb{N}} J_{n}$ where this union is pairwise disjoint.
Let $ I = [0,1] - K $ and $ J = [m,M] - \tilde{K} $, define $\Psi:I \to J$ such that: $ \Psi \left(\dfrac{1}{3}, \dfrac{2}{3}\right) = J_{1} $, where $J_{1}$ is the interval with greatest length, which is possible, since given $\varepsilon > 0$, there are only a finite amount of open sets $(a,b) \in J$ with length $b-a < M-m$. Then take $ \Psi \left(\dfrac{1}{9}, \dfrac{2}{9}\right) = J_{21} $ and $ \Psi \left(\dfrac{7}{9}, \dfrac{8}{9}\right) = J_{22} $, where $J_{21}$ and $J_{22}$ are the two greatest intervals in length such that $J_{21},J_{22} \in J - J_{1}$ (this way $\tilde{K}$ looks like $K$).
Now we define $ F $ as following: for $I_n \in I$ let $ F\vert_{I_{n}} : I_{n} \to \Psi(I_{n})$ the linear increasing function that transforms a line segment into another line segment. Then we finish this construction by saying that $ F: [0,1] \to [m, M] $ such that $ F(x) = \sup ${$F(k); k \not\in K, k \leq x $}. This is an increasing continuous bijection.
This is always possible to be built in since we take $(a,b)$ and $(c,d)$ with $b<c$ and map it to $\Psi(a,b)$ and $\Psi(c,d)$ with $\Psi(b)<\Psi(c)$, then we can finally write
$ f: \mathbb{R} \to \mathbb{R}$ such that $ f(x) = $
$x+m$, if $x<0;$
$F(x)$, if $ 0 \leq x \leq 1;$
$ x+M-1 $, if $1 < x$
I don't know how to write cases in here =)
And finally I showed that $g = f^{-1}$ is continous.
My doubts (on each part):
- How can I show that I can write $\tilde{K}$ in the same way as the Cantor set?
- How can I be sure that $J_{21}$ is to the left of $J_{21}$?
- How can I show that $ \lim\limits_{x \to a^{-}} f(x) = \lim\limits_{x \to a^{+}} f(x) \, \forall \, x \in (0,1) $?
Remark: this is from an analysis course, so I can't use any general topology arguments, so I'd like to close these gaps in my arguing.