Compact spaces and spaces of maximal complete subgraphs
Authors:
Murray Bell and John Ginsburg
Journal:
Trans. Amer. Math. Soc. 283 (1984), 329338
MSC:
Primary 54D30; Secondary 05C10, 06A10
MathSciNet review:
735426
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Abstract: We consider the space of all maximal complete subgraphs of a graph and, in particular, the space of all maximal chains of an ordered set . The main question considered is the following: Which compact spaces can be represented as for some graph or as for some ordered set ? The former are characterized as spaces which have a binary subbase for the closed sets which consists of clopen sets. We give an example to show that this does not include all zerodimensional supercompact spaces. The following negative result is obtained concerning ordered sets: Let be an uncountable discrete space and let denote the onepoint compactification of . Then there is no ordered set such that .
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 P. C. Gilmore and A. J. Hoffman, A characterization of comparability graphs and of interval graphs, Canad. J. Math. 16 (1964), 539548. MR 0175811 (31:87)
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 J. Ginsburg, I. Rival and W. Sands, Antichains and finite sets that meet all maximal chains (to appear).
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 I. Juhasz, Cardinal functions in topology, Math. Centre Tracts 34, Math. Centre, Amsterdam, 1971. MR 0340021 (49:4778)
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 J. van Mill, Supercompactness and Wallman spaces, doctoral dissertation, Free University of Amsterdam, 1977. MR 0464160 (57:4095)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198407354265
PII:
S 00029947(1984)07354265
Keywords:
Compact space,
graph,
maximal complete subgraph,
binary subbase,
ordered set,
maximal chain,
onepoint compactification
Article copyright:
© Copyright 1984
American Mathematical Society
