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proximity space (Definition)

Let $ X$ be a set. A binary relation $ \delta$ on $ P(X)$, the power set of $ X$, is called a nearness relation on $ X$ if it satisfies the following conditions: for $ A,B\in P(X)$,

  1. if $ A\cap B\ne \varnothing$, then $ A\delta B$;
  2. if $ A\delta B$, then $ A\ne \varnothing$ and $ B\ne \varnothing$;
  3. (symmetry) if $ A\delta B$, then $ B\delta A$;
  4. $ (A_1\cup A_2)\delta B$ iff $ A_1\delta B$ or $ A_2\delta B$;
  5. $ A\delta'B$ implies the existence of $ C \subseteq X$ with $ A\delta'C$ and $ (X-C)\delta'B$, where $ A\delta'B$ means $ (A,B)\notin \delta$.

If $ x,y\in X$ and $ A\subseteq X$, we write $ x\delta A$ to mean $ \lbrace x\rbrace \delta A$, and $ x\delta y$ to mean $ \lbrace x\rbrace \delta \lbrace y \rbrace$.

When $ A\delta B$, we say that $ A$ is $ \delta$-near, or just near $ B$. $ \delta$ is also called a proximity relation, or proximity for short. Condition 1 is equivalent to saying if $ A\delta'B$, then $ A\cap B=\varnothing$. 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\in A$ and $ A\delta' B$, then $ x\delta'B$.

Definition. A set $ X$ with a proximity as defined above is called a proximity space.

For any subset $ A$ of $ X$, define $ A^c=\lbrace x\in X\mid x\delta A\rbrace$. Then $ ^c$ is a closure operator on $ X$:

Proof. Clearly $ \varnothing^c=\varnothing$. Also $ A\subseteq A^c$ for any $ A\subseteq X$. To see $ A^{cc}=A^c$, assume $ x\delta A^c$, we want to show that $ x\delta A$. If not, then there is $ C\subseteq X$ such that $ x\delta'C$ and $ (X-C)\delta'A$. The second part says that if $ y\in X-C$, then $ y\delta'A$, which is equivalent to $ A^c \subseteq C$. But $ x\delta'C$, so $ x\delta'A^c$. Finally, $ x\in (A\cup B)^c$ iff $ x\delta (A\cup B)$ iff $ x\delta A$ or $ x\delta B$ iff $ x\in A^c$ or $ x\in B^c$. $ \qedsymbol$
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\in X$, $ x\delta y$ implies $ x=y$.

Examples.

  • Let $ (X,d)$ be a pseudometric space. For any $ x\in X$ and $ A\subseteq X$, define $ d(x,A):=\inf_{y\in A} d(x,y)$. Next, for $ B\subseteq X$, define $ d(A,B):=\inf_{x\in A} d(x,B)$. Finally, define $ A\delta B$ iff $ d(A,B)=0$. Then $ \delta$ 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\subseteq X$, define $ A\delta B$ iff $ A\cap B\ne\varnothing$. Then $ \delta$ 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\subseteq X$. Define $ A\delta B$ iff $ A\ne \varnothing$ and $ B\ne \varnothing$. Then $ \delta$ is also a proximity. It is called the indiscrete proximity on $ X$.

Bibliography

1
S. Willard, General Topology, Addison-Wesley, Publishing Company, 1970.
2
S.A. Naimpally, B.D. Warrack, Proximity Spaces, Cambridge University Press, 1970.



"proximity space" is owned by CWoo.
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Other names:  near, proximity, proximity relation
Also defines:  nearness relation, separated proximity space, discrete proximity, indiscrete proximity

Attachments:
proximal neighborhood (Definition) by CWoo
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Cross-references: pseudometric space, separated, induced, topological space, closure operator, subset, superset, equivalent, mean, implies, iff, symmetry, power set, binary relation
There are 68 references to this entry.

This is version 14 of proximity space, born on 2007-03-06, modified 2008-06-06.
Object id is 9037, canonical name is ProximitySpace.
Accessed 3827 times total.

Classification:
AMS MSC: 54E05 (General topology :: Spaces with richer structures :: Proximity structures and generalizations)
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