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Let be a set. A binary relation on , the power set of , is called a nearness relation on if it satisfies the following conditions: for
,
- if
, then ;
- if
, then
and
;
- (symmetry) if
, then ;
-
iff
or
;
implies the existence of
with and
, where means
.
If and
, we write to mean
, and to mean
.
When , we say that is -near, or just near . is also called a proximity relation, or proximity for short. Condition 1 is equivalent to saying if , then
. Condition 4 says that if is near , then any superset of is near . Conversely, if is not near , then no subset of is near . In particular, if and
, then .
Definition. A set with a proximity as defined above is called a proximity space.
For any subset of , define
. Then is a closure operator on :
Proof. Clearly
 . Also
 for any
 . To see
 , assume
 , we want to show that  . If not, then there is
 such that  and
 . The second part says that if  , then  , which is equivalent to
 . But  , so
 . Finally,
 iff
 iff  or  iff  or  . 
This turns 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 , implies .
Examples.
- Let
be a pseudometric space. For any and
, define
. Next, for
, define
. Finally, define iff . Then is a proximity and is a proximity space as a result.
- 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 .
- indiscrete proximity. Again,
is a non-empty set and
. Define iff
and
. Then is also a proximity. It is called the indiscrete proximity on .
- 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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