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Strong versus uniform continuity: A constructive round
Abstract
The notion of apartness has recently shown promise
as a means of lifting constructive topology from the restrictive context of
metric spaces to more general settings. Extending the point-subset apartness
axiomatised beforehand, we characterize the constructive meaning of ‘two
subsets of a given set lie apart from each other'. We propose axioms for such
apartness relations and verify them for the apartness relation associated with
an abstract uniform space. Moreover, we relate uniform continuity to strong
continuity, the natural concept for mappings between sets endowed with an
apartness structure, which says that if the images of two subsets lie apart
from each other, then so do the original subsets. Proofs are carried out with
intuitionistic logic, and most of them without the principle of countable
choice.
Mathematics Subject
Classification (2000): 54E05, 54E15, 03F60
Quaestiones Mathematicae 25 (2002), 171-190
as a means of lifting constructive topology from the restrictive context of
metric spaces to more general settings. Extending the point-subset apartness
axiomatised beforehand, we characterize the constructive meaning of ‘two
subsets of a given set lie apart from each other'. We propose axioms for such
apartness relations and verify them for the apartness relation associated with
an abstract uniform space. Moreover, we relate uniform continuity to strong
continuity, the natural concept for mappings between sets endowed with an
apartness structure, which says that if the images of two subsets lie apart
from each other, then so do the original subsets. Proofs are carried out with
intuitionistic logic, and most of them without the principle of countable
choice.
Mathematics Subject
Classification (2000): 54E05, 54E15, 03F60
Quaestiones Mathematicae 25 (2002), 171-190
