mathematical structure

From: Blake Winter (blake.winter_at_houghton.edu)
Date: 11/04/04 

Date: Thu, 4 Nov 2004 09:40:17 +0000 (UTC)

This might be more applicable to a math forum, but there should be
plenty of mathematical minded people here, so I'll ask anyway:

Is there a branch of mathematics or anyone who has studied the
following: Take a set S of points (of arbitrary cardinality). Then,
take a set G whose elements are ordered subsets of S. These sets may
be continous if S is continous, but they need not be. Call the set G
the set of paths (and note that G need not contain all possible
ordered subsets of S).

In a certain limit where S is finite and no members of G are
continous, it seems like one would get back something very like graph
theory, and with a few more restrictions on G one would actually have
graph theory. So it strikes me that perhaps this is a sort of a
generalization of graph theory.

It also seems to me like one could study objects which can't be
described by other means: consider for example if S is R^2, but the
set G consists (using the standard cartesian coordinates on R^2) only
of parametrized paths such that if dx/dt!=0, then dy/dt=0, and if
dy/dt!=0, then dx/dt=0. Such a system would be hard to describe in
anothery way.

So, does anyone know if this has been studied previously? I think it
might possibly provide a framework in which one can naturally embed
the graphs mentioned in Smolin and Markopoulou's paper
(www.arxiv.org/abs/gr-qc/0311059) into a continous manifold, without
assuming that the continous manifold arises only as an approximation.