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.