proximity space
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),
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′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′B, then xδ′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′C and (X-C)δ′A. The second part says that if y∈X-C, then yδ′A, which is equivalent to Ac⊆C. But xδ′C, so xδ′Ac. Finally, x∈(A∪B)c iff xδ(A∪B) iff xδA or xδB iff x∈Ac or x∈Bc.∎
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.
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Let (X,d) be a pseudometric space. For any x∈X and A⊆X, define d(x,A):=. Next, for , define . Finally, define iff . Then is a proximity and is a proximity space as a result.
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discrete proximity. Let be a non-empty set. For , define iff . Then so defined is a proximity on , and is called the discrete proximity on .
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indiscrete proximity. Again, is a non-empty set and . Define iff and . Then is also a proximity. It is called the indiscrete proximity on .
References
- 1 S. Willard, General Topology, Addison-Wesley, Publishing Company, 1970.
- 2 S.A. Naimpally, B.D. Warrack, Proximity Spaces, Cambridge University Press, 1970.
Title | proximity space |
Canonical name | ProximitySpace |
Date of creation | 2013-03-22 16:48:11 |
Last modified on | 2013-03-22 16:48:11 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 17 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 54E05 |
Synonym | near |
Synonym | proximity |
Synonym | proximity relation |
Defines | nearness relation |
Defines | separated proximity space |
Defines | discrete proximity |
Defines | indiscrete proximity |