In the early part of the 20th century, topological spaces were invented to capture the essence of the idea of continuity. At around the same time, other competing ideas had emerged, resulting in a variety of other “similar” types of spaces: uniform spaces and proximity spaces are the two prominent examples. These abstractions have led mathematicians to even further abstractions, in an attempt to combine all these concepts into a single construct. One such result is so-called a syntopogenous structure.
Before formally defining what a syntopogenous structure is, let us look at some of the commonalities among the three types of spaces that led to this “generalized” structure. Specifically, in all three types of sapces, we can define a transitive relation on the space such that the relation satisfies some features that are common in all three cases:
Let be a space and , we define iff
In all three cases, the relation is transitive. Furthermore, we have the following:
if , then ,
if and , then ,
if and , then ,
if , then .
By properties 2 and 6, we see that a topogenous order is a transitive antisymmetric relation.
We are now ready for the main definition.
Definition. A syntopogenous structure consists of a set and a collection of topogenous orders on such that:
if , then there is such that ,
for any , then there is such that .
Remark. The two defining conditions of a syntopogenous structure are equivalent to the following, given subsets of :
for any , there is a such that and imply ,
for any with , there is a such that for some subset of .
- 1 A. Császár, Foundations of General Topology, Macmillan, New York, 1963.
- 2 S.A. Naimpally, B.D. Warrack, Proximity Spaces, Cambridge University Press, 1970.
|Date of creation||2013-03-22 16:57:40|
|Last modified on||2013-03-22 16:57:40|
|Last modified by||CWoo (3771)|