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Order-Cauchy Completions of Rings and Vector Lattices of Continuous Functions

Published online by Cambridge University Press:  20 November 2018

F. Dashiell ,
A. Hager  and
M. Henriksen
F. Dashiell
Affiliation:
R & D Associates, Marina Del Rey, California
A. Hager
Affiliation:
Wesleyan University, Middletown, Connecticut
M. Henriksen
Affiliation:
Harvey Mudd College, Claremont, California
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This paper studies sequential order convergence and the associated completion in vector lattices of continuous functions. Such a completion for lattices C(X) is related to certain topological properties of the space X and to ring properties of C(X). The appropriate topological condition on the space X equivalent to this type of completeness for the lattice C(X) was first identified, for compact spaces X, in [6]. This condition is that every dense cozero set S in X should be C*-embedded in X (that is, all bounded continuous functions on S extend to X). We call Tychonoff spaces X with this property quasi-F spaces (since they generalize the F-spaces of [12]).

In Section 1, the notion of a completion with respect to sequential order convergence is first described in the setting of a commutative lattice group G.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 3 , 01 June 1980 , pp. 657 - 685
Copyright
Copyright © Canadian Mathematical Society 1980

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