Re: Comments on Norbert's topological extension of relational algebra

Tegiri Nenashi wrote:
> I wonder if representing topology with binary relations has been investigated.

It has. I did it.

> Naively, topology is just a binary relation, that is a bunch of ordered pairs (set_of_points, set_closure).

Well, this is not so naive:

Every topology T for a point set X can be expressed by a binary relation between X and PX, where PX is the power set of X:

There are even some alternatives:

INT \subseteq X \times PX = { (p,A) | p \in int A } -- the interior CL \subseteq X \times PX = { (p,A) | p \in cl A } -- the closure

Those relations must then satisfy certain axioms, that can be derived from the definition of interior and closure and the properties of a topology.

A special class of topologies can be expressed by a binary relation on the point sets only:

   CL \subseteq X \times X = { (p,q) | p \in cl {q} } -- the closure This is exactly my approach. Only that I use the relation transposed:

   (p,q) q \in cl{p}.
and that I have observed, that any relation R will do. CL is then the transitive and reflexove closure: CL = R^*.

Those topologies are nowadays called Alexandrov topologies.

> Assuming that algebra of binary relations is good fit for topology, then relations with
> named attributes (database relations) most likely aren't.

I don't get that point. What objections are against the names "1" and "2", (or, maybe "one" and "two").

> Here is a paper reinforcing that intuition:
> http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.135.2459&rep=rep1&type=pdf

No. This is Egenhofer's "Topological Relations" approach. It is completetly different. Received on Thu Dec 10 2015 - 19:52:40 CET