Maths - Preorders on a Finite Set

The number of topologies on a finite set is equal to the number of preorders on the set.

A preorder is like a partial order except it is not antisymmetric. That is, if x <= y and y <= x this does not imply x=y.

So a preorder has the properties of being:

• reflexive (x <= x)
• transitive (if x <= y and y <= z, then x <= z)

but not:

• anti-symmetry (if x <= y and y <= x, then x=y)
• comparability (either x <= y or y <= x or both)

Topology to Preorder

x <= y provided x belongs to every open set that contains y.

Preorder to Topology

A set T X is an open set provided there is no element in X that is less than any element in T.

Preorders for 3 Element Set

A preorder can be shown as a directed graph. Here are the corresponding topologies and preorders for a 3 element set:

Topology	Preorder
		x belongs to every open set that contains y
		A belongs to every open set that contains B A belongs to every open set that contains C
		B belongs to every open set that contains A B belongs to every open set that contains C
		C belongs to every open set that contains B C belongs to every open set that contains A
		A belongs to every open set that contains C B belongs to every open set that contains C
		A belongs to every open set that contains B C belongs to every open set that contains B
		C belongs to every open set that contains A B belongs to every open set that contains A
		A belongs to every open set that contains C
		B belongs to every open set that contains A

 

EuclideanSpace

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Where Are You

This is one of a series of pages about topology, these pages start here.

The pages about topologies on finite sets start here.

Computer Code Implementation

I have written some code to implement algebraic topology structures. For an explanation of this code see the page here.

Prerequisites

• Topology
• Set theory explained on this page.