A coverage construction of the reals and the irrationals

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...

• Type: Journal
• Year: 2007
• Where: APAL
• Authors: Harold Simmons
• Keywords: APAL 2007, Coverage Technique, Lower Section, Topological Space