The essential closure of 𝐶(𝑋)

Anderson, Marlow

doi:10.1090/S0002-9939-1979-0534378-4

The essential closure of $C(X)$
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by Marlow Anderson PDF

Proc. Amer. Math. Soc. 76 (1979), 8-10 Request permission

Abstract:

Each archimedean l-group admits a unique essential closure, which is the l-group of continuous almost finite real-valued functions on some Stonean space; thus the l-group $C(X)$ of real-valued continuous functions on a topological space X admits such an essential closure. In this note we will construct a natural embedding of $C(X)$ into its essential closure, making explicit the topological relationship between X and the appropriate Stonean space.

References

S. J. Bernau, Unique representation of Archimedean lattice groups and normal Archimedean lattice rings, Proc. London Math. Soc. (3) 15 (1965), 599–631. MR 182661, DOI 10.1112/plms/s3-15.1.599
Alain Bigard, Klaus Keimel, and Samuel Wolfenstein, Groupes et anneaux réticulés, Lecture Notes in Mathematics, Vol. 608, Springer-Verlag, Berlin-New York, 1977 (French). MR 0552653, DOI 10.1007/BFb0067004
W. W. Comfort and S. Negrepontis, The theory of ultrafilters, Die Grundlehren der mathematischen Wissenschaften, Band 211, Springer-Verlag, New York-Heidelberg, 1974. MR 0396267, DOI 10.1007/978-3-642-65780-1
Paul Conrad, The lateral completion of a lattice-ordered group, Proc. London Math. Soc. (3) 19 (1969), 444–480. MR 244125, DOI 10.1112/plms/s3-19.3.444
Paul Conrad, The essential closure of an Archimedean lattice-ordered group, Duke Math. J. 38 (1971), 151–160. MR 277457
S. Iliadis and S. Fomin, The method of concentric systems in the theory of topological spaces, Uspehi Mat. Nauk 21 (1966), no. 4 (130), 47–76 (Russian). MR 0203677