Graph spaces and -free Boolean algebras

Author:
Lutz Heindorf

Journal:
Proc. Amer. Math. Soc. **121** (1994), 657-665

MSC:
Primary 06E15; Secondary 05C99, 08A40, 54D80

DOI:
https://doi.org/10.1090/S0002-9939-1994-1246526-X

MathSciNet review:
1246526

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Abstract: Let *X* denote an arbitrary second-countable, compact, zero-dimensional space. Our main result says that *X* is a graph space, i.e., homeomorphic to the space of all complete subgraphs of a suitable graph. We first characterize graph spaces in terms of the Boolean algebras of their clopen subsets. Then it is proved that each countable Boolean algebra has the corresponding property.

As a corollary we obtain that *X* is homeomorphic to the underlying space of a subalgebra of , where **2** is the discrete two-point space and *F* any set of finitary operations on **2** such that neither the negation nor the ternary sum (addition modulo 2) belongs to the clone generated by *F*.

**[1]**M. G. Bell,*The space of complete subgraphs of a graph*, Comm. Math. Univ. Carolin.**23**(1983), 525-536. MR**677860 (84a:54050)****[2]**M. G. Bell and J. Pelant,*Continuous images of compact semilattices*, Canad. Math. Bull.**30**(1987), 109-113. MR**879879 (88c:54011)****[3]**S. Koppelberg,*General theory of Boolean algebras*, Handbook of Boolean Algebras, Vol. 1 (J. D. Monk and R. Bonnet, eds.), North-Holland, Amsterdam, 1989. MR**991565 (90k:06002)****[4]**E. L. Post,*Two-valued iterative systems of mathematical logic*, Ann. Math. Studies, vol. 5, Princeton Univ. Press, Princeton, NJ, 1941. MR**0004195 (2:337a)**

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DOI:
https://doi.org/10.1090/S0002-9939-1994-1246526-X

Article copyright:
© Copyright 1994
American Mathematical Society