Coarser uniform structure

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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: