Order topology

Remarks

When $X$ has more than one point, the open intervals $(a, b)$ together with the open rays form a basis of the order topology.

Example: on $\R$ with its usual order, the open rays generate the open intervals, so the order topology is the usual topology of $\R$.

Non-example: let $Y = \set{-1} \cup \set{1/n : n \ge 1}$ as a subset of $\R$. With the subspace topology from $\R$, the point $-1$ is isolated. In the order topology of $Y$, every open set containing $-1$ contains a ray $(\gets, 1/n)$, which contains $1/m$ for all $m > n$. So the subspace topology of $Y$ is not its order topology, although the order of $Y$ is the restriction of the order of $\R$.

Some authors also use the name for the topology generated by the same rays on an arbitrary partially ordered set.