## Bulletin of the Belgian Mathematical Society - Simon Stevin

- Bull. Belg. Math. Soc. Simon Stevin
- Volume 17, Number 5 (2010), 899-909.

### A finite axiom scheme for approach frames

Christophe Van Olmen and Stijn Verwulgen

#### Abstract

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 information

**Source**

Bull. Belg. Math. Soc. Simon Stevin, Volume 17, Number 5 (2010), 899-909.

**Dates**

First available in Project Euclid: 14 December 2010

**Permanent link to this document**

https://projecteuclid.org/euclid.bbms/1292334064

**Digital Object Identifier**

doi:10.36045/bbms/1292334064

**Mathematical Reviews number (MathSciNet)**

MR2777779

**Zentralblatt MATH identifier**

1227.06009

**Subjects**

Primary: 06D99: None of the above, but in this section 06F25: Ordered rings, algebras, modules {For ordered fields, see 12J15; see also 13J25, 16W80} 18C15: Triples (= standard construction, monad or triad), algebras for a triple, homology and derived functors for triples [See also 18Gxx] 54C40: Algebraic properties of function spaces [See also 46J10]

**Keywords**

Approach frames Eilenberg-Moore algebra

#### Citation

Van Olmen, Christophe; Verwulgen, Stijn. A finite axiom scheme for approach frames. Bull. Belg. Math. Soc. Simon Stevin 17 (2010), no. 5, 899--909. doi:10.36045/bbms/1292334064. https://projecteuclid.org/euclid.bbms/1292334064