The order topology on a totally ordered set, with the open rays as a subbasis, and its agreement with the usual topology of $\mathbb{R}$

Let $(L,\le )$ be a totally ordered set (Partial order and partially ordered set) with at least two elements. For $a\in L$ write

$L_{<a}:=\{t\in L:t<a\},L_{>a}:=\{t\in L:a<t\}$

for the open rays, and let ${\mathcal{S}}_L$ be the family of all open rays. The order topology on $L$ is ${\mathcal{T}}_{\mathrm{o}\mathrm{r}\mathrm{d}}:=⟨{\mathcal{S}}_L⟩$, the topology generated by ${\mathcal{S}}_L$ (Basis and subbasis for a topology, and the topology generated by a family of sets). Then:

1. A basis. By 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 the finite intersections of open rays form a basis for ${\mathcal{T}}_{\mathrm{o}\mathrm{r}\mathrm{d}}$, and every such intersection is $L$ itself (the empty intersection), an open ray, or an open interval $(a,b):=L_{>a}\cap L_{<b}$. So ${\mathcal{B}}_L:=\{L\}\cup {\mathcal{S}}_L\cup \{(a,b):a,b\in L\}$ is a basis for the order topology.
2. On $\mathbb{R}$ the order topology is the usual topology. Taking $L=\mathbb{R}$ with its order (Order on the reals, Ordered field), ${\mathcal{T}}_{\mathrm{o}\mathrm{r}\mathrm{d}}={\mathcal{T}}_{d_{\mathbb{R}}}$, the metric topology of $d_{\mathbb{R}}(x,y)=\mathrm{\mid }x-y\mathrm{\mid }$ (The absolute value makes $\mathbb{R}$ a metric space: $d(x,y)=\mathrm{\mid }x-y\mathrm{\mid }$ is a metric, its open balls are the intervals $(x-r,x+r)$, and it is unbounded, Metrizable space: a topological space whose topology is induced by some metric; metrizability is topological, the metric is not). The rays and intervals of claim 1 are then exactly the intervals of that shape in the sense of Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length.

Claim 2 identifies the order topology of $\mathbb{R}$ with a topology already in the library rather than introducing a second one.

• All three descriptions of $\mathbb{R}$'s topology name one collection of open sets. Claim 2 identifies the order topology with the metric topology of $d_{\mathbb{R}}$, and Which results on this page use the order of $\mathbb{R}$ and therefore have no general-topological analogue records that the metric topology and the order-native topology built earlier in this library are in turn the same collection. Nothing below uses that third description; it is named so that a reader moving between the pages knows there is one topology and not two.

• The hypothesis that $L$ has at least two elements is what keeps the rays from being useless: on a one-point set every ray is empty and the order topology is the only topology there is. Nothing else in claim 1 uses it.

• The order topology is not always metrizable, and the order alone does not decide the matter. Claim 2 is a statement about $\mathbb{R}$ and is proved from the specific fact that the balls of $d_{\mathbb{R}}$ are the bounded open intervals; no general theorem is being invoked, and none is available here.

• The Sorgenfrey line is not the order topology of the usual order on $\mathbb{R}$ (The Sorgenfrey line: $\mathbb{R}$ with the half-open intervals $[a,b)$ as a basis is strictly finer than the usual topology, is first countable, has a countable dense subset, and its sequences converge only from the right). It is generated by the half-open intervals $[a,b)$, which are not unions of open rays and open intervals, so it is strictly finer than the order topology of that order; the order it comes from is the same order, which shows that "generated by intervals" is not the same as "the order topology". Whether some other total order on $\mathbb{R}$ has the Sorgenfrey topology as its order topology is a different question, and nothing here or elsewhere in this library answers it.