coarser

The set of topologies 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 $𝒰\subset 𝒱$:

• $𝒰$ is weaker than $𝒱$

• $𝒰$ is coarser 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 equivalent to the requirement that the identity map from $(E,𝒱)$ to $(E,𝒰)$ is continuous.

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