|WikiProject Mathematics||(Rated Start-class, Low-priority)|
Note: several textbooks (Willard, Engelking, Thron) discuss proximity spaces. I can't remember if Dugundji does.--126.96.36.199 21:02, 24 Feb 2005 (UTC)
The proximity introduced (actually rediscovered) by Efrimivic on a set X, generates a completely regupar topology on it. There are a number of classes of proximities, e.g., basic proximity, Lodato proximity etc. Basic proximity generates a closure operator, Lodato proximity generates T_1 topology. Proximities are also studies through grill operators. Ref Thron, W. J., Proximity structures and grills.
I think all researchers know the definition of proximity spaces and fuzzy proximity spaces. But the thing we need to know the applications of the proximity in other branches of sciences. I hope any one know the application give us it. Prof. Abd El-Nasser G. Abd El-Rahman188.8.131.52 (talk) 19:22, 23 March 2008 (UTC)
It is absolutely not the case that all researchers know the definition of proximity spaces. This is an esoteric topic far outside the mathematical mainstream, though obviously the concepts underly a lot of mainstream mathematical concepts. In our communications with the outside world (like on Wikipedia), we should take care to not come across as too pretentious. To your main point: yes, this article needs some applications. I'm hoping that somebody can sketch applications in familiar concepts. For example, what is the proximity relation on a metric space (like or one of the spaces)? --184.108.40.206 (talk) 19:38, 17 June 2013 (UTC)
It's said "Proximity without the first axiom is called quasi-proximity."
But without this axiom we lose the dual variant of the axiom 4. It seems that in the case of quasi-proximity it should be also given the dual of the axiom 4 (and probably axiom 5):
- A δ (B∪C) ⇔ (A δ B or A δ C)
- (B∪C) δ A ⇔ (B δ A or C δ A)
- The definition in Naimpally and Warrack defines quasi-proximity, which they call P-proximity, in terms of the double version of axiom 4, but only one version of axiom 5. Spectral sequence (talk) 20:00, 24 July 2013 (UTC)
- Axiom 5 is symmetric, but Axiom 4 does need to be fixed. I will fix it now. —Toby Bartels (talk) 19:35, 13 July 2014 (UTC)
- I decided that it looked too complicated to actually add both versions (especially since this isn't needed for the non-quasi case that the article focuses on), but I put in a note that it does need to be fixed, and roughly how. (Also Axiom 2; same issue.) —Toby Bartels (talk) 19:57, 13 July 2014 (UTC)