Re: A Topological Relational Algebra in Lisp

Tegiri Nenashi wrote:
> Can we lower discussion level a little? Instead of topologies lets focus on metric spaces. Now consider multivariate relation:
>
> x y z
> -----
> 1 1 1
> 2 1 1
> 3 2 1
> 3 2 2
>
> What metric space do you expend this relation to?

I will call that relation "X".

Then, of course, the metric space will be the discrete space (X.P(X)) where P(X) is the power set of X. It will be modelled as

   (X,\emptyset)
because the empty set creates the discrete topology.

Actually, when considering databases, I am not (yet) interested very much in metric topologies for databases.

The reason is that a database relation is always finite (one might dispute that but it is a assumption a make). My model so far only considers finite database relations, and takes them as topological point sets. But a finite metric space is always discrete (where every subset of the point set is open), and, hence, quite uninteresting.

Whereas I consider the discrete topology a "default" topology for bare sets (database relations with no explicitly defined topoology) in my concept, it is a commonly known fact that finite metric topologies are always discrete. Discrete topologies are an important extreme (like the universal relation in relation theory) but not of much practical interest within my concept.

On the other hand, topological spaces that serve as a physical model of topologically organized data (say, R^3 for spaes or R^4 for space-time models) have their natural (metric) topology. So I do not neglect metric topologies. IMHO they just do not have a relational representation for interesting practical applications.

However, when spatial entities that live in some R^n (R^2 - the plane, R^3 - the Euclidean space, R^4 - the space-time, ...) are modelled and (according to my assumption) are partitioned into a a finite set of spatial entities (say, faces, edges, volumes, hyper-volumes, ...) then the set of entities immediately fits into my model (for being finite). Interestingly the entities get their topology by the same operations that produce topological result spaces from input spaces (so-called !quotient space"). Anyhow, the resulting topology for these entities (like an edge being bounded by vertices) usually does not constitute a metric sapce any more.

Maybe, however, your post might be the start of developing a concept to introduce concepts for metric topologies. Note, however, that, in order to get interesting topologies (not only the dioscrete one), the point set must be infinite.

Cheers
Norbert Received on Sun Jan 18 2015 - 00:07:11 CET