Vol. 20, No. 2, 1967

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The Dedekind completion of C(𝒳)

John Eldon Mack and Donald Glen Johnson

Vol. 20 (1967), No. 2, 231–243

The question to which this study addresses itself is the following: given a completely regular space 𝒳, is the Dedekind completion of C(𝒳) isomorphic to C(𝒴) for some space 𝒴? Here, C(𝒳) denotes the ring of continuous realvalued functions on 𝒳 under pointwise order. Affirmative answers were provided by Dilworth for the class of compact spaces in 1950 and by Weinberg for the class of countably paracompact and normal spaces in 1960. It remained an open question whether there were any spaces for which a negative answer held. In this paper, we provide a necessary and sufficient condition that the Dedekind completion of C(𝒳), for 𝒳 a realcompact space, be isomorphic to C(𝒴) for some 𝒴. Using this, we are able to provide an example of a space 𝒳 for which the Dedekind completion of C(𝒳) is not isomorphic to C(𝒴) for any space 𝒴.

Mathematical Subject Classification
Primary: 46.55
Secondary: 06.00
Received: 23 October 1964
Revised: 5 May 1965
Published: 1 February 1967
John Eldon Mack
Donald Glen Johnson