lattice of topologies

Let X be a set. Let L be the set of all topologiesMathworldPlanetmath on X. We may order L by inclusion. When 𝒯1𝒯2, we say that 𝒯2 is finer ( than 𝒯1, or that 𝒯2 refines 𝒯1.

Theorem 1.

L, ordered by inclusion, is a complete latticeMathworldPlanetmath.


Clearly L is a partially ordered setMathworldPlanetmath when ordered by . Furthermore, given any family of topologies 𝒯i on X, their intersectionMathworldPlanetmath 𝒯i also defines a topology on X. Finally, let i’s be the corresponding subbases for the 𝒯i’s and let =i. Then 𝒯 generated by is easily seen to be the supremumMathworldPlanetmathPlanetmath of the 𝒯i’s. ∎

Let L be the lattice of topologies on X. Given 𝒯iL, 𝒯:=𝒯i is called the common refinement of 𝒯i. By the proof above, this is the coarsest topology that is than each 𝒯i.

If X is non-empty with more than one element, L is also an atomic lattice. Each atom is a topology generated by one non-trivial subset of X (non-trivial being non-empty and not X). The atom has the form {,A,X}, where AX.

Remark. In general, a lattice of topologies on a set X is a sublattice of the lattice of topologies L (mentioned above) on X.

Title lattice of topologies
Canonical name LatticeOfTopologies
Date of creation 2013-03-22 16:54:42
Last modified on 2013-03-22 16:54:42
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 8
Author CWoo (3771)
Entry type Definition
Classification msc 54A10
Related topic CoarserPlanetmathPlanetmath
Defines common refinement