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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 $E$ 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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See Also: initial topology, lattice of topologies

Other names:  stronger
Also defines:  weaker, finer, refinement, expansion
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Cross-references: continuous, identity map, equivalent, mean, expressions, order, terms, inclusion, topologies
There are 110 references to this entry.

This is version 7 of coarser, born on 2002-08-13, modified 2009-02-10.
Object id is 3290, canonical name is Coarser.
Accessed 17859 times total.

Classification:
AMS MSC54-00 (General topology :: General reference works )

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