Vipul: New page: ==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 s...

2008-11-25T01:39:47Z

New page: ==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 s...

New page

==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 <math>X</math> is a set and <math>\mathcal{U}</math> and <math>\mathcal{V}</math> are two uniform structures on <math>X</math>: in other words, <math>(X,\mathcal{U})</math> is a [[uniform space]] and <math>(X,\mathcal{V})</math> is a uniform space. We say that <math>\mathcal{U}</math> is a '''coarser''' uniform structure than <math>\mathcal{V}</math> if the following equivalent conditions are satisfied:

* Any entourage in <math>\mathcal{U}</math> is in <math>\mathcal{V}</math>. In other words, <math>\mathcal{U} \subseteq \mathcal{V}</math> as subsets of <math>2^{X \times X}</math>.
* The identity map <math>(X,\mathcal{V}) \to (X,\mathcal{U})</math> is a [[uniformly continuous map]].

Coarser uniform structure - Revision history

Vipul: New page: ==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 s...