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 :
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 .
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.
|Date of creation||2013-03-22 16:48:11|
|Last modified on||2013-03-22 16:48:11|
|Last modified by||CWoo (3771)|
|Defines||separated proximity space|