I found a very interesting paper by V. Kieftenbeld and B. Löwe, "A classification of ordinal topologies" which discusses how to classify ordinals according to homeomorphism when they are given their natural order topology.
The result is based on the Cantor normal form for ordinals. Suppose the nonzero ordinal $\eta$ is written in the form (CNF) $$\eta=\omega^{\alpha_0}\cdot k_0+\dots+\omega^{\alpha_n}\cdot k_n $$ with decreasing exponents $\alpha_0>\dots>\alpha_n$ and the $k_i$ positive integers.
If the CNF has more than one term, then $\eta$ is homeomorphic to $\omega^{\alpha_0}\cdot k_0+\omega^{\alpha_n}$ (that is, remove any intermediate terms and replace the last integer with $1$). Otherwise, the CNF has just a single term. The claim is that theses cases form of complete set of ordinal representatives under homeomorphism.
Theorem: Every nonzero ordinal is homeomorphic to an ordinal of the following form:
- $\omega^\beta\cdot k$ with $k$ a positive integer,
- $\omega^\beta\cdot k+\omega^\gamma$ with $k$ a positive integer and $\gamma<\beta$.
And no two ordinals in this list are homeomorphic.
The compact ordinals in the list are the successor ordinals, of the form $\omega^\beta\cdot k+1$.
The first part of the paper is very clearly written and shows how to reduce $\alpha$ to a homeomorphic ordinal from the list above.
The second part shows that no two ordinals above are homeomorphic to each other. I am getting totally lost around Lemma 6 when they introduce their notion of "slope" and "top of slope". I think the alternative approach below would be a simpler way to go about it.
I would appreciate if someone could validate that this approach works (or give an idea what the "slope" stuff is about).
Recall the notion of Cantor-Bendixson derivatives (iterated derived sets). For a topological space $X$ and $\alpha$ ordinal: $X^{(0)}=X$; $X^{(\alpha+1)}=(X^{(\alpha)})'$ for $\alpha$ successor; $X^{(\alpha)}=\bigcap_{\delta<\alpha}X^{(\delta)}$ for $\alpha$ limit. The following is not too difficult to prove by induction:
Lemma: Given a nonzero ordinal $\eta$, its $\alpha$-th Cantor-Bendixson derivative is the set $$\eta^{(\alpha)} = \{ \omega^\alpha\cdot\zeta \mid \zeta \text{ positive ordinal with } \omega^\alpha\cdot\zeta<\eta \}.$$
Now showing non-isomorphisms among the ordinals in the list above needs checking various combinations. Lemma 5 in the paper covers basic cases.
The difficult case for me was when $\eta = \omega^\beta\cdot k+\omega^\gamma$ with $k$ a positive integer and $\gamma<\beta$. Look at $\eta^{(\alpha)}$, consisting of all positive multiples of $\omega^\alpha$ that are less than $\eta$. If $\alpha<\gamma$, there is no largest such multiple and $\eta^{(\alpha)}$ is not compact. If $\alpha=\gamma$, $\omega^\beta\cdot k$ is the largest such multiple and $\eta^{(\alpha)}$ is compact. So $\gamma$ is a topological invariant of the ordinal, namely it's the first level at which the Cantor-Bendixson derivative becomes compact.