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Topology and ordinals
Ordinals as topological spaces
Any ordinal number can be made into a topological space by endowing it with the order topology (since, being well-ordered, an ordinal is in particular totally ordered): in the absence of indication to the contrary, it is always that order topology that is meant when an ordinal is thought of as a topological space. (Note that if we are willing to accept a proper class as a topological space, then the class of all ordinals is also a topological space for the order topology.)

The set of limit points of an ordinal α is precisely the set of limit ordinals less than α. Successor ordinals (and zero) less than α are isolated points in α.
In particular, the finite ordinals and ω are discrete topological spaces, and no ordinal beyond that is discrete.

The ordinal α is compact as a topological space if and only if α is a successor ordinal.

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