Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-14T05:16:50.554Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected], [email protected]
ROLF SUABEDISSEN
Affiliation:
Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom e-mails: [email protected], [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

REFERENCES

1. Alexandroff, P., Über nulldimensionale Punktmengen, Math. Ann. 98 (1) (1928), 89–106. MR1512393Google Scholar
2. Bondy, J. A., A graph reconstructor's manual, Surveys in combinatorics, 1991 (Guildford, 1991)}, London Math. Soc. Lecture Note Ser., vol. 166 (Cambridge Univ. Press, Cambridge, 1991), 221–252. MR1161466 (93e:05071)CrossRefGoogle Scholar
3. Chandler, R. E., Hausdorff compactifications, Lecture Notes in Pure and Applied Mathematics, vol. 23 (Marcel Dekker Inc., New York, 1976). MR0515002 (58 #24191)Google Scholar
4. Engelking, R., General topology, 2nd ed., Sigma Series in Pure Mathematics, vol. 6 (Heldermann Verlag, Berlin, 1989), Translated from the Polish by the author. MR1039321 (91c:54001)Google Scholar
5. Glicksberg, I., Stone-Čech compactifications of products, Trans. Amer. Math. Soc. 90 (1959), 369–382. MR0105667 (21 #4405)Google Scholar
6. Juhász, I., Soukup, L. and Szentmiklóssy, Z., Resolvability and monotone normality, Israel J. Math. 166 (2008), 1–16. MR2430422 (2009d:54007)Google Scholar
7. Kline, J. R., A theorem concerning connected point sets, Fund. Math. 3 (1) (1922), 238–239.Google Scholar
8. Magill, K. D. Jr., N-point compactifications, Amer. Math. Mon. 72 (1965), 1075–1081. MR0185572 (32 #3036)CrossRefGoogle Scholar
9. van Mill, J., The infinite-dimensional topology of function spaces, North-Holland Mathematical Library, vol. 64 (North-Holland Publishing Co., Amsterdam, 2001). MR1851014 (2002h:57031)Google Scholar
10. Nadler, S. B. Jr, Continuum theory: An introduction, Monographs and Textbooks in Pure and Applied Mathematics, vol. 158 (Marcel Dekker, 1992).Google Scholar
11. Norden, J., Purisch, S. and Rajagopalan, M., Compact spaces of diversity two, Top. Appl. 70 (1996), 1–24.Google Scholar
12. Sierpinski, W., Sur une propriété topologique des ensembles dénombrables dense en soi, Fund. Math. 1 (1920), 11–16.Google Scholar
13. Ward, A. J., The topological characterisation of an open linear interval, Proc. London Math. Soc. 41 (1) (1936), 191–198.CrossRefGoogle Scholar
14. Watson, S., The construction of topological spaces: Planks and resolutions, in (Hušek, M. and van Mill, J., Editors) Recent progress in general topology (Prague, 1991) (North-Holland, Amsterdam, 1992), 673–757. MR1229141Google Scholar
15. Whyburn, G. T., Concerning the cut points of continua, Trans. Amer. Math. Soc. 30 (3) (1928), 597–609. MR1501448CrossRefGoogle Scholar