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Reconstructing compact metrizable spaces

Authors: Paul Gartside, Max F. Pitz and Rolf Suabedissen
Journal: Proc. Amer. Math. Soc. 145 (2017), 429-443
MSC (2010): Primary 54E45; Secondary 05C60, 54B05, 54D35
Published electronically: July 22, 2016
MathSciNet review: 3565393
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Abstract: The deck, $ \mathcal {D}(X)$, of a topological space $ X$ is the set $ \mathcal {D}(X) =\{[X \setminus \{x\}]:x \in X\}$, where $ [Y]$ denotes the homeomorphism class of $ Y$. A space $ X$ is (topologically) reconstructible if whenever $ \mathcal {D}(Z)=\mathcal 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 $ \langle B_n^x:n \in \mathbb{N}\rangle $ of pairwise disjoint clopen subsets converging to $ x$ such that $ B_n^x$ and $ B_n^y$ 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 $ G_\delta $. For $ h$-homogeneous spaces, this condition is sufficient for non-reconstruction. A wide variety of spaces with a dense $ G_\delta $ set of $ 1$-point components is presented, some reconstructible and others not reconstructible.

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Additional Information

Paul Gartside
Affiliation: The Dietrich School of Arts and Sciences, 301 Thackeray Hall, Pittsburgh, Pennsylvania 15260

Max F. Pitz
Affiliation: Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom
Address at time of publication: Department of Mathematics, University of Hamburg, Bundesstraße 55 (Geomatikum), 20146 Hamburg, Germany

Rolf Suabedissen
Affiliation: Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom

Keywords: Reconstruction conjecture, topological reconstruction, finite compactifications, universal sequence
Received by editor(s): October 19, 2015
Received by editor(s) in revised form: March 10, 2016
Published electronically: July 22, 2016
Additional Notes: The second author is the corresponding author
This research formed part of the second author’s thesis at the University of Oxford (2015), supported by an EPSRC studentship.
Communicated by: Mirna Džamonja
Article copyright: © Copyright 2016 American Mathematical Society

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