uniform proximity is a proximity
In this entry, we want to show that a uniform proximity is, as expected, a proximity.
First, the following equivalent characterizations of a uniform proximity is useful:
Lemma 1.
Let X be a uniform space with uniformity U, and A,B are subsets of X. Denote U[A] the image of A under U∈U:
{b∈X∣(a,b)∈U for some a∈A}. |
The following are equivalent:
-
1.
(A×B)∩U≠∅ for all U∈𝒰
-
2.
U[A]∩U[B]≠∅ for all U∈𝒰
-
3.
U[A]∩B≠∅ for all U∈𝒰
If we define AδB iff the pair A,B satisfy any one of the above conditions for all U∈𝒰, we call δ the uniform proximity.
Proof.
(1⇒2) Suppose (a,b)∈(A×B)∩U. Then b∈U[A]. Since U is reflexive, (b,b)∈U, or b∈U[B]. This means b∈U[A]∩U[B].
(2⇒3) For any U∈𝒰, we can find V∈𝒰 such that V∘V⊆U. So V=V∘Δ⊆V∘V⊆W, where Δ is the diagonal relation (since V is reflexive). Set W=V∩V-1. By assumption, there is c∈W[A]∩W[B] (and hence c∈U[A]∩U[B] as well). This means (a,c),(b,c)∈W for some a∈A and b∈B. Since W is symmetric
, (c,b)∈W⊆V, so that (a,b)=(a,c)∘(c,b)∈V⊆U. This means that b∈U[A]. As a result, U[A]∩B≠∅.
(3⇒1) If b∈U[A]∩B, then there is a∈A such that (a,b)∈U, or (A×B)∩U≠∅. ∎
We want to prove the following:
Proposition 1.
Proof.
We verify each of the axioms of a proximity relation:
-
1.
if , then :
pick , then since the diagonal relation for all .
-
2.
if , then and :
If , then for every , since no is empty, there is such that , or and .
-
3.
(symmetry) if , then :
If , then there is for every , so , which implies , or .
-
4.
iff or :
Since ,
iff iff -
5.
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 |
---|---|
Canonical name | UniformProximityIsAProximity |
Date of creation | 2013-03-22 18:07:21 |
Last modified on | 2013-03-22 18:07:21 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 7 |
Author | CWoo (3771) |
Entry type | Derivation |
Classification | msc 54E17 |
Classification | msc 54E05 |
Classification | msc 54E15 |