# 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 $\mathcal{U}\subset\mathcal{V}$:

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

• $\mathcal{U}$ is 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.

 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