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 š’Æ1āŠ†š’Æ2, we say that š’Æ2 is finer (http://planetmath.org/Finer) than š’Æ1, or that š’Æ2 refines š’Æ1.

Theorem 1.

L, ordered by inclusion, is a complete latticeMathworldPlanetmath.

Proof.

Clearly L is a partially ordered setMathworldPlanetmath when ordered by āŠ†. Furthermore, given any family of topologies š’Æi on X, their intersectionMathworldPlanetmath ā‹‚š’Æi also defines a topology on X. Finally, let ā„¬iā€™s be the corresponding subbases for the š’Æiā€™s and let ā„¬=ā‹ƒā„¬i. Then š’Æ generated by ā„¬ is easily seen to be the supremumMathworldPlanetmathPlanetmath of the š’Æiā€™s. āˆŽ

Let L be the lattice of topologies on X. Given š’ÆiāˆˆL, š’Æ:=ā‹š’Æi is called the common refinement of š’Æi. By the proof above, this is the coarsest topology that is than each š’Æi.

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 {āˆ…,A,X}, where āˆ…āŠ‚AāŠ‚X.

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