Wednesday, February 14, 2018

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.

Tuesday, January 30, 2018

Subgraphs of connected graphs

Given a connected graph as well a subset of the vertices of the connected graph, we can form a subgraph of the connected graph containing only the vertices in that subset. This subgraph need not be connected itself, or even non empty as it could consist of disconnected points. The interesting property of the subgraph then is actually its metric rather then its adjacency relation. We can form a metric on the subgraph by determining the distance between each point in the parent graph based upon the shortest path metric.

Sunday, December 31, 2017

Metric properties of graphs

Given a connected graph, we can form a metric space from the graph in which the distance between any two points is determined by the shortest path between them. The shortest path between two points need not be unique, unless the graph is a geodetic graph. This metric space is always going to be a discrete space, and a very particular type of discrete space which is generated by the unit distances between its points. The most interesting metric property of any vertex in a graph is its eccentricity which is the greatest distance between the vertex and any other point. The radius and the diameter of a graph are distance related properties determined by the minimum and the maximum eccentricity respectively.

Thursday, August 24, 2017

Geodetic graphs

Geodetic graphs have the particular property that each pair of points has a unique shortest path between them, or geodesic. Trees are a special case of geodetic graphs in which each pair of points has a unique path in general between them. Block graphs are sometimes called clique trees, they include trees and also happen to be the chordal geodetic graphs.

Monday, April 3, 2017

Partition logic

The logic of set partitions is a fundamental part of the analysis of ordering relations. The set partitions necessarily form a lattice which has certain properties corresponding to its join and meet operations. At the same time, the elements themselves form a Moore family depending upon which direction you intend to take the ordering. The Moore family then necessarily comes with a corresponding closure operation.