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Boundedness Properties in Function-Lattices

Published online by Cambridge University Press:  20 November 2018

M. H. Stone
M. H. Stone*
Affiliation:
University of Chicago
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The continuous real functions on a topological space X are partially ordered in a natural way by putting ƒ ≦ g if and only if ƒ(x)≦ g(x) for all x in X. With respect to this partial ordering these functions constitute a lattice, the lattice operations ∪ and ∩ being defined by the relations (ƒ ∪ g) (x) = max (ƒ(x), g(x)) ƒ ∩ g )(x) = min (ƒ(x), g(x)). The lattice character of any partially ordered system merely expresses the existence of least upper and greatest lower bounds for any finite set of elements in the system. Many partially ordered systems enjoy much stronger boundedness properties than these: for example, every subset with an upper bound may have a least upper bound, as in the case of the real number system.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 1 , Issue 2 , 01 April 1949 , pp. 176 - 186
Copyright
Copyright © Canadian Mathematical Society 1949

References

1 The results of this paper, for the case where X is compact, were announced in general terms and without proof in the Proceedings of the National Academy of Sciences, vol. 26 (1940), 280-283; and important applications were indicated there. Full discussions for the case where X is a completely regular space satisfying the category condition (C) of Sec. 2 were presented in lectures at Harvard, Brown, Chicago, and the Universidad del Literal (Rosario, Argentina) in 1942 and 1943. As referee, Professor I. Halperin pointed out that it would be desirable to isolate and minimize the rôle played by the condition (C). His detailed suggestions to this end have been incorporated in Sec. 2.

2 The proof is supplied by observing that ϕ X ƒ is a function of Baire. For implications of this theorem, see Stone, footnote 4.

3 Nakano, H., Proc. Imp. Acad. Tokyo, vol. 17 (1941), 308–310 Google Scholar

4 See Stone, Bull. Amer. Math. Soc,vol. 44 (1938), 807-816; Trans. Amer. Math. Soc, vol. 40 (1936), 37-111, and vol. 41 (1937), 375-481. In Sec. 7 of the Bulletin paper (which is a brief general survey) the union of any non-void subclass of a Boolean ring is defined. The property of ℵ-additivity is the property that the union of every subclass of at most ℵ members exists. In the topological representation (Theorem 67 of the first Transactions paper, and Theorems 1, 2, 4 of the second) this property is equivalent to the following: the union G of any family of at mostfcℵ closed-and-open sets is contained in a smallest closed-andopen set, necessarily the closure of G.