The theory of approach spaces has set the context in which numerical topological concepts exist. The successful interaction between frames and topology on the one hand and the search for a good notion of sobriety in the context of approach theory on the other hand was the motivation to develop a theory of approach frames.
The original definition of approach frames was given in terms of an implicitly defined set of equations. In this work, we describe a subset of this by a finite axiom scheme (of only six types of equations) which implies all the equations originally involved and hence provides a substantial simplification of the definition of approach frames. Furthermore we show that the category of approach frames is the Eilenberg-Moore category for the monad determined by the functor which takes each approach frame to the set of its regular functions.
@article{1292334064,
author = {Van Olmen, Christophe and Verwulgen, Stijn},
title = {A finite axiom scheme for approach frames},
journal = {Bull. Belg. Math. Soc. Simon Stevin},
volume = {17},
number = {1},
year = {2010},
pages = { 899-909},
language = {en},
url = {http://dml.mathdoc.fr/item/1292334064}
}
Van Olmen, Christophe; Verwulgen, Stijn. A finite axiom scheme for approach frames. Bull. Belg. Math. Soc. Simon Stevin, Tome 17 (2010) no. 1, pp. 899-909. http://gdmltest.u-ga.fr/item/1292334064/