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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.
Defines:
weaker, finer, refinement, expansion
Related:
InitialTopology, LatticeOfTopologies
Synonym:
stronger
Type of Math Object:
Definition
Major Section:
Reference
Groups audience:
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