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A TOPOLOGICAL VARIATION OF THE RECONSTRUCTION CONJECTURE

Part of: Spaces with richer structures Graph theory Generalities Basic constructions

Published online by Cambridge University Press:  10 June 2016

MAX F. PITZ  and
ROLF SUABEDISSEN
MAX F. PITZ
Affiliation:
Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom e-mails: pitz@maths.ox.ac.uk, suabedis@maths.ox.ac.uk
ROLF SUABEDISSEN
Affiliation:
Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom e-mails: pitz@maths.ox.ac.uk, suabedis@maths.ox.ac.uk
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Abstract

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This paper investigates topological reconstruction, related to the reconstruction conjecture in graph theory. We ask whether the homeomorphism types of subspaces of a space X which are obtained by deleting singletons determine X uniquely up to homeomorphism. If the question can be answered affirmatively, such a space is called reconstructible. We prove that in various cases topological properties can be reconstructed. As main result we find that familiar spaces such as the reals ℝ, the rationals ℚ and the irrationals ℙ are reconstructible, as well as spaces occurring as Stone–Čech compactifications. Moreover, some non-reconstructible spaces are discovered, amongst them the Cantor set C.

MSC classification

Primary: 54B99: None of the above, but in this section
Secondary: 05C60: Isomorphism problems (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.) 54B05: Subspaces 54A05: Topological spaces and generalizations (closure spaces, etc.) 54E35: Metric spaces, metrizability
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 59 , Issue 1 , January 2017 , pp. 221 - 235
Copyright
Copyright © Glasgow Mathematical Journal Trust 2016 

References

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