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Two Constructive Embedding-Extension Theorems with Applications to Continuity Principles
 

Summary: Two Constructive Embedding-Extension Theorems
with Applications to Continuity Principles
and to Banach-Mazur Computability
Andrej Bauer
Alex Simpson
December 19, 2003
Abstract
We prove two embedding and extension theorems in the context of the constructive theory
of metric spaces. The first states that Cantor space embeds in any inhabited complete sepa-
rable metric space (CSM) without isolated points, X, in such a way that every sequentially
continuous function from Cantor space to Z extends to a sequentially continuous function
from X to R. The second asserts an analogous property for Baire space relative to any inhab-
ited locally non-compact CSM. Both results rely on having careful constructive formulations
of the concepts involved.
As a first application, we derive new relationships between "continuity principles" asserting
that all functions between specified metric spaces are pointwise continuous. In particular, we
give conditions that imply the failure of the continuity principle "all functions from X to
R are continuous", when X is an inhabited CSM without isolated points, and when X is
an inhabited locally non-compact CSM. One situation in which the latter case applies is in
models based on "domain realizability", in which the failure of the continuity principle for any

  

Source: Andrews, Peter B. - Department of Mathematical Sciences, Carnegie Mellon University

 

Collections: Mathematics