Coarser uniform structure

Definition

Symbol-free definition

Given two uniform structures on a set, we say that the first structure is coarser than the second if the following equivalent conditions are satisfied:

• Any entourage for the first uniform structure is an entourage for the second uniform structure.
• The identity map is uniformly continuous from the second uniform structure to the first.

Definition with symbols

Suppose $X$ is a set and $\mathcal{U}$ and $\mathcal{V}$ are two uniform structures on $X$: in other words, $(X,\mathcal{U})$ is a uniform space and $(X,\mathcal{V})$ is a uniform space. We say that $\mathcal{U}$ is a coarser uniform structure than $\mathcal{V}$ if the following equivalent conditions are satisfied:

• Any entourage in $\mathcal{U}$ is in $\mathcal{V}$. In other words, $\mathcal{U}\subseteq \mathcal{V}$ as subsets of $2^{X\times X}$.
• The identity map $(X,\mathcal{V})\to (X,\mathcal{U})$ is a uniformly continuous map.