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Original Paper
On local non-compactness in recursive mathematics
Article first published online: 24 JUL 2006
DOI: 10.1002/malq.200510036
Copyright © 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1521-3870/asset/cover.gif?v=1&s=3df32dbed8d9153ac6eafe1eaa786854b9b48345 "Mathematical Logic Quarterly")
Additional Information
How to Cite
Simonsen, J. G. (2006), On local non-compactness in recursive mathematics. Mathematical Logic Quarterly, 52: 323–330. doi: 10.1002/malq.200510036
Publication History
- Issue published online: 24 JUL 2006
- Article first published online: 24 JUL 2006
- Manuscript Accepted: 8 MAY 2006
- Manuscript Revised: 5 MAY 2006
- Manuscript Received: 1 NOV 2005
- Abstract
- References
- Cited By
Keywords:
- Constructive mathematics;
- recursive mathematics;
- metric spaces;
- compactness;
- recursion theory
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)