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 $\mathcal{U}$ and $\mathcal{V}$ be two topologies defined on a set $E$. All of the following expressions mean that $\mathcal{U}\subset \mathcal{V}$:

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$\mathcal{U}$ is weaker than $\mathcal{V}$

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$\mathcal{U}$ is coarser^{} than $\mathcal{V}$

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$\mathcal{V}$ is finer than $\mathcal{U}$

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$\mathcal{V}$ is a refinement of $\mathcal{U}$

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$\mathcal{V}$ is an expansion of $\mathcal{U}$
It is worth noting that this condition is equivalent^{} to the requirement that the identity map from $(E,\mathcal{V})$ to $(E,\mathcal{U})$ is continuous^{}.
Title  coarser 
Canonical name  Coarser 
Date of creation  20130322 12:56:03 
Last modified on  20130322 12:56:03 
Owner  rspuzio (6075) 
Last modified by  rspuzio (6075) 
Numerical id  10 
Author  rspuzio (6075) 
Entry type  Definition 
Classification  msc 5400 
Synonym  stronger 
Related topic  InitialTopology 
Related topic  LatticeOfTopologies 
Defines  weaker 
Defines  finer 
Defines  refinement 
Defines  expansion 