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On local non-compactness in recursive mathematics

✍ Scribed by Jakob G. Simonsen

Publisher
John Wiley and Sons
Year
2006
Tongue
English
Weight
132 KB
Volume
52
Category
Article
ISSN
0044-3050

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✦ Synopsis

Abstract

A metric space is said to be locally non‐compact if every neighborhood contains a sequence that is eventually bounded away from every element of the space, hence contains no accumulation point. We show within recursive mathematics that a nonvoid complete metric space is locally non‐compact iff it is without isolated points.

The result has an interesting consequence in computable analysis: If a complete metric space has a computable witness that it is without isolated points, then every neighborhood contains a computable sequence that is eventually computably bounded away from every computable element of the space. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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