What is the point of endowing an ordinal with the order topology?

Question

Let $\alpha$ be an ordinal. Then $(\alpha,\mathcal{O}_\alpha)$ is a topological space if we let $\mathcal{O}_\alpha$ be generated by the basis $$\mathcal{B}:=\{\emptyset\}\cup \bigcup_{\beta<\beta'\le\alpha \text{ Ordinals}} \{\{\gamma \text{ Ordinal} \mid \beta<\gamma<\beta'\}\} $$

, which is simply the order topology on the totally ordered set $(\alpha,\in|_{\alpha\times\alpha})$.

For somebody who is familiar with order topologies, this might help with building intuition as to how the ordinals behave.

However, for somebody who isn't familiar with topologies beyond the basic definitions, is there any use one gets out of this topology that he should know about?

This was an exercise in a book, so it probably won't use this topology anymore. In the exercise I proofed that the topology is Hausdorff, that precisely the limit points aren't isolated, and that $(\alpha,\mathcal{O}_\alpha)$ is compact iff $\alpha$ is not a limit ordinal.

One application to topology is described in the following: Countable compact spaces as ordinals AND How many compact Hausdorff spaces are there of a given cardinality? AND A note on countable compact spaces by V. Kannan (1974). — Dave L. Renfro, Feb 14 '25 at 01:10

score 4 · Accepted Answer · answered Feb 14 '25 at 01:13

As mentioned, ordinals provide massive amount of examples of topological spaces in general topology. For instance, the Dowker space originally constructed by Rudin was created as a subspace of the box product of ordinal spaces $\omega_n$.

For another use of ordinal spaces, one can use them to show that given any compact Hausdorff space $X$, there exists a locally compact Hausdorff space $Y$ such that $\beta Y\setminus Y$ is homeomorphic to $X$, where $\beta Y$ is the Stone-Cech compactification of $Y$, the largest compactification. To do this you simply let $Y = X\times \omega_\alpha$ where $\alpha$ is large enough, so that $\beta Y$ becomes $X\times (\omega_\alpha+1)$.

score 3 · Answer 2 · answered Feb 14 '25 at 00:58

In addition to ordinal topologies providing interesting examples in general topology (e.g. $\omega_1$ is sequentially compact but not compact), the topology on the ordinals does come up in set theory. For instance, closed and unbounded subsets of regular cardinals (usually called clubs) are very important (c.f. Jech Chapter 8, or pretty much any reference on set theory), and of course "closed" here means in the sense of the order topology.

What is the point of endowing an ordinal with the order topology?

2 Answers2