The order topology on an ordinal, with the half-open intervals $(\alpha, \beta]$ and the initial segments $[0, \beta]$ as a basis

Let $\gamma$ be an ordinal (Ordinal (von Neumann)). Since $\gamma$ is the set of ordinals below it and $\xi < \eta$ means $\xi \in \eta$, the following two families of subsets of $\gamma$ are defined for $\beta \in \gamma$ and $\alpha \in \gamma$:

$[0,\beta] \;:=\; \{\, \xi \in \gamma : \xi \le \beta \,\} \;=\; \beta^{+}, \qquad (\alpha,\beta] \;:=\; \{\, \xi \in \gamma : \alpha < \xi \le \beta \,\} \;=\; \beta^{+} \setminus \alpha^{+} .$

Both identifications are immediate: $\beta^{+} = \beta \cup \{\beta\}$ is the set of ordinals $\le \beta$, and it is a subset of $\gamma$ because $\gamma$ is transitive and $\beta \in \gamma$ (Ordinal (von Neumann), Basic closure properties of ordinals).

Put

$\mathcal{B}_\gamma \;:=\; \{\, [0,\beta] : \beta \in \gamma \,\} \;\cup\; \{\, (\alpha,\beta] : \alpha, \beta \in \gamma,\ \alpha < \beta \,\} .$

$\mathcal{B}_\gamma$ is a basis for a unique topology on $\gamma$ (A family is a basis for a unique topology iff it covers the set and every point of an intersection of two members lies in a member inside that intersection; finite intersections of any subbasis form a basis, Basis and subbasis for a topology, and the topology generated by a family of sets), and that topology is the order topology on $\gamma$. The obligation is discharged here.

(B1), covering. If $\xi \in \gamma$ then $\xi \in [0,\xi] \in \mathcal{B}_\gamma$, so $\bigcup \mathcal{B}_\gamma = \gamma$. For $\gamma = 0 = \varnothing$ the family is empty and $\bigcup \varnothing = \varnothing = \gamma$, so (B1) holds there too.

(B2), intersections. By trichotomy of the ordinals (Trichotomy and well-ordering of the ordinals) any two ordinals have a maximum and a minimum, namely the larger and the smaller of the two, and for $\alpha_1, \alpha_2, \beta_1, \beta_2 \in \gamma$:

• $[0,\beta_1] \cap [0,\beta_2] = [0, \min\{\beta_1,\beta_2\}]$;
• $[0,\beta_1] \cap (\alpha_2,\beta_2] = (\alpha_2, \min\{\beta_1,\beta_2\}]$ when $\alpha_2 < \min\{\beta_1,\beta_2\}$, and $\varnothing$ otherwise;
• $(\alpha_1,\beta_1] \cap (\alpha_2,\beta_2] = (\max\{\alpha_1,\alpha_2\},\ \min\{\beta_1,\beta_2\}]$ when $\max\{\alpha_1,\alpha_2\} < \min\{\beta_1,\beta_2\}$, and $\varnothing$ otherwise.

In each case the intersection is either a member of $\mathcal{B}_\gamma$ or empty, and in the empty case (B2) is vacuous, having no point to test. So (B2) holds, and A family is a basis for a unique topology iff it covers the set and every point of an intersection of two members lies in a member inside that intersection; finite intersections of any subbasis form a basis supplies the topology and its uniqueness.

This definition is for ordinals only, and it says so. The general order topology of a linearly ordered set takes the open intervals, together with the initial and final rays, as a basis. For an ordinal that family is the wrong one: a successor $\beta^{+}$ has an immediate predecessor, so the smallest open interval around it is already $\{\beta^{+}\}$, but no interval of the form $(\alpha,\eta)$ isolates $0$, and the initial segments must be supplied separately. The family $\mathcal{B}_\gamma$ above is exactly the general order basis for an ordinal, rewritten so that no case analysis is needed; nothing here claims to define the order topology of an arbitrary linearly ordered set, and no statement on this page is about such a set.

Isolated and non-isolated points. Every ordinal is $0$, a successor, or a limit (Successor and limit ordinals). If $\xi = 0$ then $\{\xi\} = [0,0]$ is basic open; if $\xi = \alpha^{+}$ then $\{\xi\} = (\alpha, \xi]$ is basic open; so every non-limit point of $\gamma$ is isolated. If $\xi$ is a limit ordinal then every basic set containing $\xi$ contains some $(\alpha,\xi]$ with $\alpha < \xi$, and $\alpha^{+} < \xi$ because $\xi$ is a limit, so $\alpha^{+}$ is a second point of that basic set; hence a limit point of $\gamma$ is not isolated. In particular $\omega$, the least limit ordinal ($\omega$ is the least limit ordinal), is the unique non-isolated point of $\omega + 1$, and every ordinal $\gamma \le \omega$ carries the discrete topology (The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies).

• The basis members are clopen, and that is proved as the next item; it is the single fact that makes ordinal spaces easy to place in the separation hierarchy, since a clopen basis gives regularity at once.

• $\gamma$ is a set of ordinals and also a space. The notations $[0,\beta]$ and $(\alpha,\beta]$ are relative to the ambient $\gamma$: the same symbols in a larger ordinal denote larger sets. Where two ordinals are in play the ambient one is named.

• Nothing here needs any choice principle. Every fact used above is a theorem of ZF (Basic closure properties of ordinals, Trichotomy and well-ordering of the ordinals).

• This is the ordinal instance of a general construction. The order topology of an arbitrary linearly ordered set is now defined elsewhere in the library (The order topology of a linearly ordered set, with the open rays as a subbasis; order-convex sets, order-density, the least upper bound property, and linear continua), by open intervals together with the initial and final rays. $\mathcal{B}_\gamma$ above is not that construction applied verbatim to $\gamma$ — the note two Remarks up already explains why a raw open-interval basis is the wrong family for an ordinal — but it generates the same topology: every basic open set of the general construction is a union of members of $\mathcal{B}_\gamma$ and conversely, since both bases are generated by the same order relation on the same underlying set. So this definition is the ordinal special case of that one, restated in a form that needs no case analysis, not a second, competing notion.