Poincare Project: The Standard Topology for Ordered Sets

One common way of defining a topology is to take a set, add some structure to that set, define a collection of subsets that meet some criteria in that structure and then use that collection as a basis for the open sets.

Although we didn't have the vocabulary to accurately describe it in those terms, that's what we did previously with the topology of a metric space. A metric space, recall, adds to a set the structure of a distance function. From this, we can define the collection of open balls. This collection can then form the basis for the other open sets in a topology.

Here is another example. Take a set X and add to it the structure of a total ordering. A total ordering is a relationship < such that

• for any a, b, c in X: a < b and b < c implies a < c
• for any a, b in X: a < b or b < a or a = b

In other words, a set with a total ordering is a set whose elements can be sorted.

Now define an open interval (a, b) to be the subset of X such that, for each element x, a < x and x < b.

The open intervals form the basis for a topology. So a total ordering on a set defines a particular topology. While other topologies are possible, the one based on the open intervals is referred to as the standard topology for the ordering or the order topology.

The real numbers, being a totally ordered set, has an order topology. While other topologies can be defined on the real numbers (as long as the rules for open sets are followed), the order topology is the most natural and consistent with one's intuitions about how the real numbers work.

UPDATE: next post

The original post was in the category: poincare_project but I'm still in the process of migrating categories over.