fundamental system of entourages
To see that each uniform space has a fundamental system of entourages, define
There is a useful equivalent condition for being a fundamental system of entourages. Let be a nonempty family of subsets of . Then is a fundamental system of entourages of a uniformity on if and only if it the following axioms.
(B1) If , , then contains an element of .
(B2) Each element of contains the diagonal .
(B3) For any , the inverse relation of contains an element of .
Suppose is a fundamental system of entourages for uniformities and . Then . To see this, suppose . Since is a fundamental system of entourages for , there is some element such that . But , so . Hence by applying the fact that is closed under taking supersets we may conclude that . So if is a fundamental system of entourages, it is a fundamental system for a unique uniformity . Thus it makes sense to call the uniformity generated by the fundamental system .
- 1 Nicolas Bourbaki, Elements of Mathematics: General Topology: Part 1, Hermann, 1966.
|Title||fundamental system of entourages|
|Date of creation||2013-03-22 16:29:55|
|Last modified on||2013-03-22 16:29:55|
|Last modified by||mps (409)|
|Defines||uniformity generated by|