uniformities on a set form a complete lattice
Let be a set. Let denote the collection of uniformities on . The coarsest uniformity on is , and the finest is the discrete uniformity:
(B1) Let , . Each of and is a finite intersection of elements of the , so their intersection is as well. Hence .
(B2) Every element of is a finite intersection of subsets of containing . So every element of contains the diagonal.
(B4) Let . Again suppose with and . Then there exist and such that and . The set is in , and since is a subset of both and , it is a subset of .
The fundamental system generates a uniformity . By construction, is an upper bound of the . But any upper bound of the would have to contain all finite intersections of elements of the . So . ∎
Let be a set and let be a family of uniform spaces. Then for any family of functions , there is a coarsest uniformity on making all the uniformly continous.
The coarsest uniformity making a family of functions uniformly continuous is called the initial uniformity or weak uniformity.
- 1 Nicolas Bourbaki, Elements of Mathematics: General Topology: Part 1, Hermann, 1966.
|Title||uniformities on a set form a complete lattice|
|Date of creation||2013-03-22 16:30:46|
|Last modified on||2013-03-22 16:30:46|
|Last modified by||mps (409)|