Notation lost in Space

One of the more commonly used terms in linear algebra, and other areas that make heavy use of linear algebra, is that of a subspace. This term, I’ve been teaching a course on linear algebra, and we’ve gotten to the chapter on subspaces. Frequently, I end up saying X is a subspace of Y or some such thing. It seems to me that much of mathematical notation comes about because certain phrases and words are repeated so often that it greatly simplifies [1] writing if there is notation for them.

I’ve said this so frequently in class, and had to write it, that I’m beginning to wish that there were notation for subspace, yet, to the best of my knowledge, there isn’t. “X is a subspace of Y” is analogous to both “x is less than y” or “A is contained in B”/”A is a subset of B”. These three relations all fit in the framework of partially ordered sets. There is notation for the latter two. Why haven’t we come up with a corresponding notation for subspace? Or have we, and I just didn’t get the memo? (If I ever teach an algebra course, I’ll be complaining about the lack of notation for subgroup as well.)

[1] Well, it makes for less time writing, anyway. On the other hand, students don’t much care for new notation, so maybe it’s better to do without it.

Asymmetric metric

A metric on a space X is a function from to R satisfying the following properties

1. 
2. if and only if
3. for all
4. for all

The standard example of this is probably the usual Euclidean metric on Euclidean space. Now it is easy to find “real life” examples in which the fourth axiom doesn’t hold, but the other three do. For example, pick any city that has at least one one-way street. Then the distance from point A to point B along that one way street will be different from B to A, unless perhaps the one way street is a circle and A and B are on opposite points on the circle. For any other pair of points, the distances are not equal.

A discrete example is a strongly connected digraph (the graph must be strongly connected, for if not, then there is a pair of vertices a and b such that there is no path from a to b, and thus the distance between the two is infinite, which contradicts that fact that the distance must be a real number).

A continuous example is a circle in which it is only possible to travel clockwise.

A function which satisfies all but condition 4 in the definition of metric is called a quasimetric (The word quasimetric may have different meanings, depending on the context). Elsewhere it is simply referred to as an asymmetric metric, or asymmetric distance, (these meaning of these terms are also inconsistent). Whatever you call it, there seems to be little of substance on the matter. The term quasimetric seems to yield the best results on Google.

Is there a candidate for a “standard” example of a (non-trivial) quasimetric space? Is there a standard quasimetric on Euclidean space?

Is there a relevant source that covers the non-trivialities of quasimetrics?

Are there any other real-life examples of quasi-metrics besides cities with one way streets?

New Graph Theory Book

For years, one of the standard references in graph theory was “Graph Theory with Applications” by J.A. Bondy and U.S.R Murty, first published in 1976. More recently other texts have played that role. On November 30, 2007 a follow up to Bondy and Murty’s book, entitled simply “Graph Theory”, was published.

The blog for the book can be found here.

The Springer page for the book can be found here.

I haven’t been able to find a review of the book. If you know of one, please let me know.