Compact spaces and spaces of maximal complete subgraphs

Authors:
Murray Bell and John Ginsburg

Journal:
Trans. Amer. Math. Soc. **283** (1984), 329-338

MSC:
Primary 54D30; Secondary 05C10, 06A10

DOI:
https://doi.org/10.1090/S0002-9947-1984-0735426-5

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 zero-dimensional supercompact spaces. The following negative result is obtained concerning ordered sets: Let be an uncountable discrete space and let denote the one-point compactification of . Then there is no ordered set such that .

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1984-0735426-5

Keywords:
Compact space,
graph,
maximal complete subgraph,
binary subbase,
ordered set,
maximal chain,
one-point compactification

Article copyright:
© Copyright 1984
American Mathematical Society