Coarser uniform structure
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 is a set and and are two uniform structures on : in other words, is a uniform space and is a uniform space. We say that is a coarser uniform structure than if the following equivalent conditions are satisfied:
- Any entourage in is in . In other words, as subsets of .
- The identity map is a uniformly continuous map.