Proximity and Nearness
Proximity is introduced when trying to axiomatize the concept of being in some sense “near” that may hold from a set to another set. Far better known are point-to-set notions that characterize being element of a neighborhood or of an open set. The first concept of proximity was described in 1908 by Frigyes Riesz and then ignored. Others to be mentioned for having worked on such ideas include V. A. Efremovič in 1934 and A. N. Wallace in 1940. More recently, we found some work in Naimpally and Warrack (Proximity Spaces, Cambridge University Press, 1970), Vakarelov et al. (J Appl Non-Class Log 12:527–559, 2002), Bennett and Düntsch (Axioms, Algebras and Topology. In Marco Aiello, Ian E. Pratt-Hartmann, and Johan F.A.K. van Bentham, editors, Handbook of Spatial Logics, pages 99–159.Springer, 2007).
- [BD07]Brandon Bennett and Ivo Düntsch. Axioms, Algebras and Topology. In Marco Aiello, Ian E. Pratt-Hartmann, and Johan F.A.K. van Bentham, editors, Handbook of Spatial Logics, pages 99–159. Springer-Verlag, 2007.Google Scholar
- [BStV01]Douglas Bridges, Peter Schuster, and Luminiţa Vîţă. Apartness, topology, and uniformity: a constructive view, 2001. Submission found in the internet.Google Scholar
- [DL12]Ivo Düntsch and Sanjiang Li. Extension Properties of Boolean Contact Algebras. In Timothy G. Griffin and Wolfram Kahl, editors, RAMICS 2012, number 7560 in Lect. Notes in Comput. Sci., pages 342–356. Springer-Verlag, 2012.Google Scholar
- [GW14]Manas Ghosh and Michael Winter. Refinements of the RCC25 Composition Table. In Peter Höfner, Peter Jipsen, Wolfram Kahl, and Martin E. Müller, editors, RAMICS, number 8428 in Lect. Notes in Comput. Sci., pages 379–394. Springer-Verlag, 2014.Google Scholar
- [NW70]Somashekhar Amrith Naimpally and Brian D. Warrack. Proximity Spaces. Cambridge University Press, 1970.Google Scholar