A coverage construction of the reals and the irrationals

13 years 4 months ago

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I modify the standard coverage construction of the reals to obtain the irrationals. However, this causes a jump in ordinal complexity from ω + 1 to Ω. The coverage technique has its origins in the generation of Gabriel and Grothendieck topologies. Later the technique was modiﬁed for use in point-free topology, and for the generation of genuine topological spaces. In the setting the technique takes the form of a rather simple kind of relation r ⊢ U between elements r of a poset S and lower section U of S. We think of each r ∈ S as a name for a basic open set, and each U ∈ LS (the family of all lower section of S) as a name for an arbitrary open set of the space under construction. We wish to read this relation as ‘the basic open set named by r is included in the open set named by U’ (which more or less tells us what the general properties of ⊢ should be). Each such relation is speciﬁed by certain postulated primitive instances. We may think of S together with these p...

Harold Simmons

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