Why is the interval topology defined not in terms of open rays but in terms of complements of closed rays?

Question

In the literature, see e.g. Frink (1942) or Erné (1980), the interval topology on a partially ordered set $(X,\leq)$ is commonly defined as the topology generated by complements of closed rays. A closed ray is a set $\{w\in X:w\leq x\}$ or $\{y\in X:y\geq x\}$, where $x\in X$.

If $X$ is the set of real numbers then an equivalent but simpler definition of the interval topology is the topology generated by open rays, i.e., sets $\{w\in X:w < x\}$ or $\{y\in X:y > x\}$, where $x\in X$.

But, while defining an open interval in terms of open rays is a simpler than defining it in terms of complements of closed rays, this apparently is not the preferred way to go. Apparently the topology generated by complements of closed rays is nicer to work with than the topology generated by open rays. Why is that?

Answer 1

In a totally ordered $(X,\le)$ complements of closed rays agree with open rays. In a non-totally partially ordered $(X,\le)$ this is no longer true. In general we have $R(x,>) \subsetneqq X - R(x,\le)$. This indicates that the open rays will generate a different topology than the complements of closed rays. There is a priori no reason why one of these topologies should be "superior". However, the purpose of topologizing partially ordered sets is to establish a correspondence between the properties of $X$ as a partially ordered set and the properties of $X$ as a topological space. See for example

Wolk, E. S. "Order-compatible topologies on a partially ordered set." Proceedings of the American Mathematical Society 9.4 (1958): 524-529

which gives arguments for the "traditional" choice.

Answer 2

I would say that the property of the interval topology of a lattice is better than the topology generated by open intervals. For example, for a poset, the interval topology is $T_1$.

Frink-Birkhoff theorem: For a lattice $L$, the interval topology is compact iff $L$ is a complete lattice.