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	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