We want to talk about the “distance” between objects in a set.

Our inspiration is the notion of distance in the plane (see Plane Distance) or in space (see Space Distance). The objects are points and the distance between them is the length of the line segment joining them. We note a few properties of this notion of distance:

- The distance between any two distinct objects is not zero.
- The distance between any two objects does not depend on the order in which we consider them.
- The distance between two objects is no larger than the sum of the distances of each with any third object

The first observation is natural: if two points are not the same, then they are some distance apart. In other words, the line segment between them has length.

The second observation is natural: the line segment connecting two points does not depend on the order specifying the points. This observation justifies the word “between.” If it were not the case, then we should use different words, and be careful to speak of the distance “from” a first point “to” a second point.

The third property is a non-obvious property of distance in the plane. It says, in other words, that the length of any side of a triangle is no larger than the sum of the lengths of the two other sides. With experience in geometry, the observation may become natural. But it does not seem to be superficially so.

A more muddled but superficially natural justification for our concern with third observation is that it says something about the transitivity of closeness. Two objects are close if their distance is small. Small is a relative concept, and needs some standard of comparison. Let us fix two points, take the distance between them, and call it a unit. We call two objects close with respect to our unit if their distance is less than a unit.

In this language, the third observation says that if we know two objects are each half of a unit distance from a third object, then the two objects are close (their distance is less than a unit). We might call this third object the reference object. Here, then, is the usefulness of the third property: we can infer closeness of two objects if we know their distance to a reference object.