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.