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Reference: Gartside, P, Pitz, MF and Suabedissen, R et al., (2016). Reconstructing compact metrizable spaces. Proceedings of the American Mathematical Society, 145 (1), 429-443.Citable link to this page:


Reconstructing compact metrizable spaces

Abstract: The deck, D(X), of a topological space X is the set D(X) = {[X\{x}]: x ∈ X}, where [Y] denotes the homeomorphism class of Y. A space X is (topologically) reconstructible if whenever D(Z) = D(X), then Z is homeomorphic to X. It is known that every (metrizable) continuum is reconstructible, whereas the Cantor set is non-reconstructible. The main result of this paper characterises the non-reconstructible compact metrizable spaces as precisely those where for each point x there is a sequence of pairwise disjoint clopen subsets converging to x such that Bxn and Byn are homeomorphic for each n and all x and y. In a non-reconstructible compact metrizable space the set of 1-point components forms a dense Gδ. For h-homogeneous spaces, this condition is sufficient for non-reconstruction. A wide variety of spaces with a dense Gδ set of 1-point components is presented, some reconstructible and others not reconstructible.

Publication status:PublishedPeer Review status:Peer reviewedVersion:Accepted manuscriptDate of acceptance:2016-05-03Notes:© 2016 American Mathematical Society. This is the accepted manuscript version of the article. The final version is available online from the American Mathematical Society at: [10.1090/proc/13270]

Bibliographic Details

Publisher: American Mathematical Society

Publisher Website:

Journal: Proceedings of the American Mathematical Societysee more from them

Publication Website:

Volume: 145

Issue: 1

Extent: 429-443

Issue Date: 2016-07-22



Eissn: 1088-6826

Issn: 0002-9939

Uuid: uuid:69a78f6d-d401-41fd-9ebb-e9b4e50e8d87

Urn: uri:69a78f6d-d401-41fd-9ebb-e9b4e50e8d87

Pubs-id: pubs:571283 Item Description

Type: journal-article;

Version: Accepted manuscriptKeywords: reconstruction conjecture topological reconstruction finite compactifications universal sequence


Author: Gartside, P - - - Pitz, MF - Oxford, MPLS, Mathematical Institute fundingEngineering and Physical Sciences Research Council grant



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