The Nonstraddling Lemma and a new look at the Heine-Borel Theorem

Spain, P. G. (2017) The Nonstraddling Lemma and a new look at the Heine-Borel Theorem. Working Paper. Researchgate.

Abstract

Any family of intervals in the real line determines a linking equivalence relation on its union. The equivalence classes are order convex so are therefore themselves intervals, partitioning the union of the original family. If one starts from a cover of a bounded closed interval by open intervals one can then apply the Nonstraddling Lemma, a result of the utmost simplicity, to clinch the proof of the Heine-Borel Theorem. The Structure Theorem for Open Sets emerges naturally from this discussion. The natural setting for this scheme of proof is in the wider context of complete linearly ordered spaces where the generalisation of the Heine-Borel Theorem is known as the Haar-König Theorem.

• Item Type: Research Reports or Papers (Working Paper)
• Status: Published
• Refereed: No
• Glasgow Author(s) Enlighten ID: Spain, Dr Philip
• Authors: Spain, P. G.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Publisher: Researchgate
• Copyright Holders: Copyright © 2017 The Author
• Publisher Policy: Reproduced with the permission of the Author

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