The set of topologiesMathworldPlanetmathPlanetmath which can be defined on a set is partially ordered under inclusion. Below, we list several synonymous terms which are used to refer to this order. Let 𝒰 and 𝒱 be two topologies defined on a set E. All of the following expressions mean that 𝒰𝒱:

  • 𝒰 is weaker than 𝒱

  • 𝒰 is coarserPlanetmathPlanetmath than 𝒱

  • 𝒱 is finer than 𝒰

  • 𝒱 is a refinement of 𝒰

  • 𝒱 is an expansion of 𝒰

It is worth noting that this condition is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to the requirement that the identity map from (E,𝒱) to (E,𝒰) is continuousMathworldPlanetmathPlanetmath.

Title coarser
Canonical name Coarser
Date of creation 2013-03-22 12:56:03
Last modified on 2013-03-22 12:56:03
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 10
Author rspuzio (6075)
Entry type Definition
Classification msc 54-00
Synonym stronger
Related topic InitialTopology
Related topic LatticeOfTopologies
Defines weaker
Defines finer
Defines refinement
Defines expansion