Summary: Two Constructive Embedding-Extension Theorems
with Applications to Continuity Principles
and to Banach-Mazur Computability
December 19, 2003
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 separable 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 inhabited 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 inhabited locally non-compact CSM, X, generalizes a result
previously obtained by EscardŽo and Streicher in the special case X = C[0, 1].