PlanetMath: coarser

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coarser

(Definition)

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 . All of the following expressions mean that $\mathcal{U} \subset \mathcal{V}$ :

$\mathcal{U}$ is weaker than $\mathcal{V}$
$\mathcal{U}$ is coarser than $\mathcal{V}$
$\mathcal{V}$ is finer than $\mathcal{U}$
$\mathcal{V}$ is a refinement of $\mathcal{U}$
$\mathcal{V}$ 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.

"coarser" is owned by rspuzio. [ full author list (3) | owner history (1) ]

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Also defines:

weaker, finer, refinement, expansion

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Cross-references: continuous, identity map, equivalent, mean, expressions, order, terms, inclusion, topologies
There are 23 references to this entry.

This is version 6 of coarser, born on 2002-08-13, modified 2007-06-12.
Object id is 3290, canonical name is Coarser.
Accessed 14424 times total.

Classification:

AMS MSC:

54-00 (General topology :: General reference works )

Pending Errata and Addenda

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