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Transactions of the American Mathematical Society

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Semirings and $ T\sb{1}$ compactifications. I

Author: Douglas Harris
Journal: Trans. Amer. Math. Soc. 188 (1974), 241-258
MSC: Primary 54D35; Secondary 54C40
MathSciNet review: 0365492
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Abstract: With each infinite cardinal $ {\omega _\mu }$ is associated a topological semiring $ {{\mathbf{F}}_\mu }$, whose underlying space is finite complement topology on the set of all ordinals less than $ {\omega _\mu }$, and whose operations are the natural sum and natural product defined by Hessenberg. The theory of the semirings $ {C_\mu }(X)$ of maps from a space X into $ {{\mathbf{F}}_\mu }$ is developed in close analogy with the theory of the ring $ C(X)$ of continuous real-valued 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 $ {C_\mu }(X)$ for sufficiently large $ {\omega _\mu }$ will be the Wallman compactification of X. The classes of $ {\omega _\mu }$-entire and $ {\omega _\mu }$-total spaces, which are respectively analogues of realcompact and pseudocompact spaces, are examined, and an $ {\omega _\mu }$-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 $ {\omega _\mu }$-entire extensions. The semiring of functions of compact-small support is considered, and shown to be related to the locally compact-small spaces in the same way that the ring of functions of compact support is related to locally compact spaces.

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