Semirings and compactifications. I
Author:
Douglas Harris
Journal:
Trans. Amer. Math. Soc. 188 (1974), 241258
MSC:
Primary 54D35; Secondary 54C40
MathSciNet review:
0365492
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Abstract: With each infinite cardinal is associated a topological semiring , whose underlying space is finite complement topology on the set of all ordinals less than , and whose operations are the natural sum and natural product defined by Hessenberg. The theory of the semirings of maps from a space X into is developed in close analogy with the theory of the ring of continuous realvalued functions; the analogy is not on the surface alone, but may be pursued in great detail. With each semiring a structure space is associated; the structure space of for sufficiently large will be the Wallman compactification of X. The classes of entire and total spaces, which are respectively analogues of realcompact and pseudocompact spaces, are examined, and an entire extension analogous to the Hewitt realcompactification is constructed with the property (not possessed by the Wallman compactification) that every map between spaces has a unique extension to their entire extensions. The semiring of functions of compactsmall support is considered, and shown to be related to the locally compactsmall spaces in the same way that the ring of functions of compact support is related to locally compact spaces.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197403654924
PII:
S 00029947(1974)03654924
Article copyright:
© Copyright 1974
American Mathematical Society
