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proximity space∗
CWoo†
2013-03-21 22:14:37
Let X be a set. A binary relation δ on P (X), the power set of X, is called a
nearness relation on X if it satisfies the following conditions: for A, B ∈ P (X),
1. if A ∩ B 6= ∅, then AδB;
2. if AδB, then A 6= ∅ and B 6= ∅;
3. (symmetry) if AδB, then BδA;
4. (A1 ∪ A2 )δB iff A1 δB or A2 δB;
5. Aδ 0 B implies the existence of C ⊆ X with Aδ 0 C and (X − C)δ 0 B, where
Aδ 0 B means (A, B) ∈
/ δ.
If x, y ∈ X and A ⊆ X, we write xδA to mean {x}δA, and xδy to mean
{x}δ{y}.
When AδB, we say that A is δ-near, or just near B. δ is also called a
proximity relation, or proximity for short. Condition 1 is equivalent to saying if
Aδ 0 B, then A ∩ B = ∅. Condition 4 says that if A is near B, then any superset
of A is near B. Conversely, if A is not near B, then no subset of A is near B.
In particular, if x ∈ A and Aδ 0 B, then xδ 0 B.
Definition. A set X with a proximity as defined above is called a proximity
space.
For any subset A of X, define Ac = {x ∈ X | xδA}. Then c is a closure
operator on X:
Proof. Clearly ∅c = ∅. Also A ⊆ Ac for any A ⊆ X. To see Acc = Ac , assume
xδAc , we want to show that xδA. If not, then there is C ⊆ X such that xδ 0 C
and (X − C)δ 0 A. The second part says that if y ∈ X − C, then yδ 0 A, which is
equivalent to Ac ⊆ C. But xδ 0 C, so xδ 0 Ac . Finally, x ∈ (A ∪ B)c iff xδ(A ∪ B)
iff xδA or xδB iff x ∈ Ac or x ∈ B c .
∗ hProximitySpacei created: h2013-03-21i by: hCWooi version: h39037i Privacy setting:
h1i hDefinitioni h54E05i
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This turns X into a topological space. Thus any proximity space is a topological space induced by the closure operator defined above.
A proximity space is said to be separated if for any x, y ∈ X, xδy implies
x = y.
Examples.
• Let (X, d) be a pseudometric space. For any x ∈ X and A ⊆ X, define d(x, A) := inf y∈A d(x, y). Next, for B ⊆ X, define d(A, B) :=
inf x∈A d(x, B). Finally, define AδB iff d(A, B) = 0. Then δ is a proximity and (X, d) is a proximity space as a result.
• discrete proximity. Let X be a non-empty set. For A, B ⊆ X, define AδB
iff A ∩ B 6= ∅. Then δ so defined is a proximity on X, and is called the
discrete proximity on X.
• indiscrete proximity. Again, X is a non-empty set and A, B ⊆ X. Define
AδB iff A 6= ∅ and B 6= ∅. Then δ is also a proximity. It is called the
indiscrete proximity on X.
References
[1] S. Willard, General Topology, Addison-Wesley, Publishing Company, 1970.
[2] S.A. Naimpally, B.D. Warrack, Proximity Spaces, Cambridge University
Press, 1970.
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