Coarser uniform structure

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.