TITLE: Willard 17R; uncountably many compact subsets of real line
QUESTION [5 upvotes]: I am self-studying topology and came across question 17R of Willard's General Topology.

17R. Compact subsets of $\mathbb{R}$
There are uncountably many nonhomeomorphic compact subsets of $\mathbb{R}$. [Use ordinals.]

The discussions I found which are similar (e.g. Uncountably many non-homeomorphic compact subsets of the circle) use what seems to be more advanced stuff ("Cantor-Bendixson rank", for example).
I guess the hint suggest us to look at $\Omega=[0,\omega_1]$, where $\omega_1$ is the first uncountable ordinal. I can do the following:

Every countable ordinal embeds into $\mathbb{R}$. This is more-or-less straightforward induction.

So the problem boils down in proving that there are uncountably many non-homeomorphic countable ordinals. It is also clear that if $\alpha$ is an infinite ordinal and $\beta$ is the largest limit ordinal $\leq\alpha$, then the compacts $[0,\alpha]$ and $[0,\beta]$ are homeomorphic.
I can also prove that there are uncountably many countable limit ordinals, but some of these are homeomorphic to each other (e.g. $\omega^2+\omega$ and $\omega^2$).
I would appreciate help, using not much more than basic facts about $\omega_1$ (as it is introduced in Willard's book).

REPLY [3 votes]: Using Eric Wofsey’s lemma we can explicitly construct an uncountable family of pairwise non-homeomorphic countable, compact spaces that embed in $\Bbb R$. They are the same ones that we’d get using a Cantor-Bendixson approach, but without all of that machinery.
If $X$ is a compact space that can be embedded in $[0,1]$, let $X^*$ be the one-point compactification of $\omega\times X$, where $\omega$ has the discrete topology; it’s not hard to show that $X^*$ can also be embedded in $[0,1]$. Use Eric Wofsey’s lemma to show that if $X$ is countable, then $X^*$ is not homeomorphic to $X$.
Now let $X_0$ be the compact ordinal space $\omega+1$. Given $X_\alpha$ for some ordinal $\alpha$, let $X_{\alpha+1}=(X_\alpha)^*$. If $\alpha$ is a countable limit ordinal, and $X_\eta$ has been defined for each $\eta<\alpha$, let $Y_\alpha=\bigsqcup_{\eta<\alpha}X_\eta$, and let $X_\alpha=(Y_\alpha)^*$; since $\alpha$ is countable, $X_\alpha$ can be embedded in $[0,1]$, and we can continue the recursive construction to get for each $\alpha<\omega_1$ a countable, compact space $X_\alpha$ that embeds in $[0,1]$.
Suppose that $\alpha<\beta<\omega_1$; then $X_{\alpha+1}\subseteq X_\beta$, and $X_{\alpha+1}$ contains disjoint copies of $X_\alpha$, so the lemma ensures that $X_\beta$ cannot be homeomorphic to $X_\alpha$, and $\{X_\alpha:\alpha<\omega_1\}$ is therefore an uncountable family of mutually non-homeomorphic countable, compact spaces that can be embedded in $[0,1]$.