syntopogenous structure
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 socalled 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 $X$ be a space and $A,B\subseteq X$, we define $A\le B$ iff

•
(topological) $A\subseteq {B}^{\circ}$, the interior of $B$.

•
(uniform) $U[A]\subseteq B$ for some entourage $U$. $U[A]$ is a uniform neighborhood of $A$.

•
(proximity) $A{\delta}^{\prime}(XB)$, where $\delta $ is the proximity relation, and ${\delta}^{\prime}$ is its complement^{}.
In all three cases, the relation is transitive^{}. Furthermore, we have the following:

1.
$\mathrm{\varnothing}\le \mathrm{\varnothing}$,

2.
$X\le X$,

3.
if $A\le B$, then $A\subseteq B$,

4.
if $A\le B$ and $C\le D$, then $A\cap C\le B\cap D$,

5.
if $A\le B$ and $C\le D$, then $A\cup C\le B\cup D$,

6.
if $A\subseteq B\le C\subseteq D$, then $A\le D$.
Definition. Let $X$ be a set. A topogenous order $\le $ on $X$ is a binary relation on $P(X)$, the powerset of $X$, satisfying the six properties above.
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 $X$ and a collection^{} $\mathcal{S}$ of topogenous orders on $X$ such that:

•
if ${R}_{1},{R}_{2}\in \mathcal{S}$, then there is $R\in \mathcal{S}$ such that ${R}_{1}\cap {R}_{2}\subseteq R$,

•
for any $R\in \mathcal{S}$, then there is $S\in \mathcal{S}$ such that $R\subseteq S\circ S$.
Remark. The two defining conditions of a syntopogenous structure $(X,\mathcal{S})$ are equivalent^{} to the following, given subsets $A,B$ of $X$:

•
for any ${\le}_{1},{\le}_{2}\in \mathcal{S}$, there is a $\le \in \mathcal{S}$ such that $A{\le}_{1}B$ and $A{\le}_{2}B$ imply $A\le B$,

•
for any ${\le}_{1}\in \mathcal{S}$ with $A{\le}_{1}B$, there is a ${\le}_{2}\in \mathcal{S}$ such that $A{\le}_{2}C{\le}_{2}B$ for some subset $C$ of $X$.
References
 1 A. Császár, Foundations of General Topology, Macmillan, New York, 1963.
 2 S.A. Naimpally, B.D. Warrack, Proximity Spaces, Cambridge University Press, 1970.
Title  syntopogenous structure 

Canonical name  SyntopogenousStructure 
Date of creation  20130322 16:57:40 
Last modified on  20130322 16:57:40 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  7 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 54A15 
Defines  topogenous order 