Extending strongly continuous functions between apartness spaces

A natural extension theorem for strongly continuous mappings, the morphisms in the category of apartness spaces, is proved constructively. The author thanks Douglas Bridges and the anonymous referee for their useful suggestions, and the FoRST New Zealand for supporting her as a Postdoctoral Research Fellow (contract number UOCX0215) during the writing of this paper.     

Open subspaces of locally compact metric spaces

Although classically every open subspace of a locally compact space is also locally compact, constructively this is not generally true. This paper provides a locally compact remetrization for an open set in a compact metric space and constructs a one-point compactification. MSC: 54D45, 03F60, 03F65.     

Boolean Algebras and Distributive Lattices Treated Constructively

Some aspects of the theory of Boolean algebras and distributive lattices–in particular, the Stone Representation Theorems and the properties of filters and ideals–are analyzed in a constructive setting.     

Constructive notions of equicontinuity

In the informal setting of Bishop-style constructive reverse mathematics we discuss the connection between the antithesis of Specker’s theorem, Ishihara’s principle BD-N, and various types of equicontinuity. In particular, we prove that the implication from pointwise equicontinuity to uniform sequential equicontinuity is equivalent to the antithesis of Specker’s theorem; and that, for a family of functions on a separable metric space, the implication from uniform sequential equicontinuity to uniform equicontinuity is equivalent to BD-N.      

Finitary sequence spaces

This paper studies the metric structure of the space H_r of absolutely summable sequences of real numbers with at most r nonzero terms. H_r is complete, and is located and nowhere dense in the space of all absolutely summable sequences. Totally bounded and compact subspaces of H_r are characterized, and large classes of located, totally bounded, compact, and locally compact subspaces are constructed. The methods used are constructive in the strict sense. MSC: 03F65, 54E50.     

Walker's Cancellation Theorem

Walker's cancellation theorem says that, if B ⊕ Z is isomorphic to C ⊕ Z in the category of abelian groups, then B is isomorphic to C. We construct an example in a diagram category of abelian groups where the theorem fails. As a consequence, the original theorem does not have a constructive proof even if B and C are subgroups of the free abelian group on two generators. Both of these results contrast with a group whose endomorphism ring has stable range one, which allows a constructive proof of cancellation and also a proof in any diagram category.