syntopogenous structure

In the early part of the 20th century, topological spacesMathworldPlanetmath were invented to capture the essence of the idea of continuity. At around the same time, other competing ideas had emerged, resulting in a varietyMathworldPlanetmath 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” structureMathworldPlanetmath. Specifically, in all three types of sapces, we can define a transitive relation on the space such that the relationMathworldPlanetmath satisfies some features that are common in all three cases:

Let X be a space and A,BX, we define AB iff

  • (topological) AB, the interior of B.

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

  • (proximity) Aδ(X-B), where δ is the proximity relation, and δ is its complementPlanetmathPlanetmath.

In all three cases, the relation is transitiveMathworldPlanetmathPlanetmathPlanetmathPlanetmath. Furthermore, we have the following:

  1. 1.


  2. 2.


  3. 3.

    if AB, then AB,

  4. 4.

    if AB and CD, then ACBD,

  5. 5.

    if AB and CD, then ACBD,

  6. 6.

    if ABCD, then AD.

Definition. Let X be a set. A topogenous order 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 collectionMathworldPlanetmath 𝒮 of topogenous orders on X such that:

  • if R1,R2𝒮, then there is R𝒮 such that R1R2R,

  • for any R𝒮, then there is S𝒮 such that RSS.

Remark. The two defining conditions of a syntopogenous structure (X,𝒮) are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to the following, given subsets A,B of X:

  • for any 1,2𝒮, there is a 𝒮 such that A1B and A2B imply AB,

  • for any 1𝒮 with A1B, there is a 2𝒮 such that A2C2B for some subset C of X.


  • 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 2013-03-22 16:57:40
Last modified on 2013-03-22 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