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The pseudocompactness of [0.1] is equivalent to the uniform continuity theorem

Published online by Cambridge University Press:  12 March 2014

Douglas Bridges
Affiliation:
Department of Mathematics & Statistics, University of Canterbury, Private Bag 4800, Christchurch, New Zealand. E-mail: d.bridges@math.canterbury.ac.nz
Hannes Diener
Affiliation:
Department of Mathematics & Statistics, University of Canterbury, Private Bag 4800, Christchurch, New Zealand. E-mail: H.Diener@math.canterbury.ac.nz

Abstract

We prove constructively that, in order to derive the uniform continuity theorem for pointwise continuous mappings from a compact metric space into a metric space, it is necessary and sufficient to prove any of a number of equivalent conditions, such as that every pointwise continuous mapping of [0, 1] into ℝ is bounded. The proofs are analytic, making no use of, for example, fan-theoretic ideas.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2007

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References

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