uniform proximity is a proximity
Let be a uniform space with uniformity , and are subsets of . Denote the image of under :
The following are equivalent:
If we define iff the pair satisfy any one of the above conditions for all , we call the uniform proximity.
() Suppose . Then . Since is reflexive, , or . This means .
() For any , we can find such that . So , where is the diagonal relation (since is reflexive). Set . By assumption, there is (and hence as well). This means for some and . Since is symmetric, , so that . This means that . As a result, .
() If , then there is such that , or . ∎
We want to prove the following:
The binary relation on defined by
is a proximity on .
We verify each of the axioms of a proximity relation:
if , then :
pick , then since the diagonal relation for all .
if , then and :
If , then for every , since no is empty, there is such that , or and .
(symmetry) if , then :
If , then there is for every , so , which implies , or .
iff or :
implies the existence of with and , where means .
First note that is symmetric because is. By assumption, there is such that (second equivalent characterization of uniform proximity from lemma above). Set . Then . By the third equivalent condition of uniform proximity, . Likewise, , so , or .
This shows that is a proximity on . ∎
|Title||uniform proximity is a proximity|
|Date of creation||2013-03-22 18:07:21|
|Last modified on||2013-03-22 18:07:21|
|Last modified by||CWoo (3771)|