Bulletin of the Belgian Mathematical Society - Simon Stevin

A finite axiom scheme for approach frames

Christophe Van Olmen and Stijn Verwulgen

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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.

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

First available in Project Euclid: 14 December 2010

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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]

Approach frames Eilenberg-Moore algebra


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

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