Order topology

Given a totally ordered set, we can form an open topology on that set from the set of open rays consisting of all the points that are either strictly greater or strictly less then a given point. The order topology also contains the open intervals of the set. The first concept that can be derived from the order topology is that of an isolated point. An isolated point is a singleton set of the order topology. A discrete total order consists entirely of isolated points. A point is considered to be near isolated if it always contains an isolated point in any open set containing it.

A scattered topological space is strictly near isolated. These scattered topological points consist of both isolated points, and limits of isolated points. Scattered topologies include discrete topologies as a special case. In this sense, they are somewhat of a generalization of discrete topologies. Points are often defined by the existence of a topological subspace that contains them. A scattered point is a point that is contained within some scattered topology. Scattered total orders are defined as total orders with a scattered topology.

A topological space that contains no isolated points is dense in itself, which makes it relatively less restricted then these other types of topological spaces. The other spaces are defined based upon forbidding dense subspaces. A space can include dense subspaces as well as isolated points and be neither type of topology. A point can be characterized based upon rather it is contained in a dense in itself. Dense total orders are defined as total orders with a dense in itself topology. The real numbers themselves have a dense in itself topology.

A metric space is defined based upon a totally ordered set of distances. The character that a metric space can take is determined by the order topology of its set of distances. If a metric space has a discrete set of distances then it is necessarily going to be a discrete metric. For example, the path metric on a graph uses only integer distances so it will necessarily only form a discrete metric. In the same sense, if a scattered set of distances is used, then the metric space will necessarily be scattered as well. As a result, being either discrete or scattered is transferred from the order topology of the distances to the metric space. In this sense, the order topology is perhaps the most fundamental concept in the theory of metric spaces.

It is useful to define a metric space associated with a given partial order. In a locally finite order this can be defined based upon the path metric of the covering graph. If the order has a different topology, however, then it is necessary to define a different type of metric space on it. In particular, if an order has a dense topology then the path metric may no longer suffice and it will be necessary to define some other concept of distance between points. So a dense metric can be created so that it can be associated with the dense order.