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Locally non-compact spaces and continuity principles
Bauer, Andrej ; Simpson, Alex
We give a constructive proof that Baire space embeds in any inhabited locally non-compact complete separable metric space, ▫$X$▫, in such a way that every sequentially continuous function from Baire ...space to ▫$\mathbb Z$▫ extends to a function from ▫$X$▫ to ▫$\mathbb R$▫. As an application, we show that, in the presence of certain choice and continuity principles, the statement "all functions from ▫$X$▫ to ▫$\mathbb R$▫ are continuous" is false. This generalizes a result previously obtained by Escardó and Streicher in the context of "domain realizability", for the special case ▫$X={\mathcal C}[0,1]$▫.
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