Abstract
We characterize Riesz spaces C(X) of real-valued continuous functions on a topological (Tychonoff) space X without using the Yosida representation theorem. The approach is elementary by using a simple description of zero-sets and a kind of local uniform completeness avoiding inversion closeness.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Anderson F.W.: Approximation in systems of real-valued continuous functions. Trans. Am. Math. Soc. 103, 249–271 (1962)
Brandenburg H., Mysior A.: Short proof of an internal characterization of complete regularity. Canad. Math. Bull. 27(4), 461–462 (1984)
Feldman W.A., Porter J.F.: The order topology for function lattices and realcompactness. Internat. J. Math. Math. Sci. 4(2), 289–304 (1981)
Henriksen M., Johnson D.J.: On the structure of a class of Archimedean lattice-ordered algebras. Fund. Math. 50, 73–94 (1961)
Hušek M., Pulgarín A.: C(X) as a real ℓ-group. Topol. Appl. 157(8), 1454–1459 (2010)
Hušek M., Pulgarín A.: C(X)-objects in the category of semi-affine lattices. Appl. Categ. Struct. 19(2), 439–454 (2011)
Hušek, M., Pulgarín, A.: General approach to characterizations of C(X). Topol. Appl. 159, 1603–1612 (2012)
Jensen G.A.: A note on complete separation in the Stone topology. Proc. Am. Math. Soc. 21, 113–116 (1969)
Johnson D.G., Mandelker M.: Separating chains in topological spaces. J. Lond. Math. Soc. 4(2), 510–512 (1971)
Kerstan J.: Eine Cahrakterisierung der vollständing regulären Räume. Math. Nachr. 17, 27–46 (1958)
Luxembourg, W.A.J., Zaanen, A.C.: Riesz Spaces I. North-Holland, Amsterdam (1981)
Montalvo F., Pulgarín A., Requejo B.: Zero-separating algebras of continuous functions. Topol. Appl. 154(10), 2135–2141 (2007)
Montalvo F., Pulgarín A., Requejo B.: Riesz spaces of real continuous functions. Positivity 14(3), 473–480 (2010)
Xiong H.Y.: A characterization of Riesz spaces wich are Riesz isomorphic to C(X) for some completely regular space X. Indag. Math. 51(1), 87–95 (1989)
Yosida K.: On the representation of the vector lattices. Proc. Imp. Akad. Tokyo 18, 339–342 (1942)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hušek, M., Pulgarín, A. On characterizing Riesz spaces C(X) without Yosida representation. Positivity 17, 515–524 (2013). https://doi.org/10.1007/s11117-012-0185-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11117-012-0185-5
Keywords
- Archimedean Riesz space
- Locally uniformly complete
- Real-valued continuous functions
- Weak order unit
- Zero-separation