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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
DOI: https://doi.org/10.1090/S0002-9947-1974-0365492-4
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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  • [BU] N. Bourbaki, General topology. Part I, Addison-Wesley, Reading, Mass., 1966. MR 34 #5044b.
  • [FGL] N. Fine, L. Gillman and J. Lambek, Rings of quotients of rings of functions, McGill Univ. Press, Montreal, Que., 1966. MR 34 #635. MR 0200747 (34:635)
  • [GJ] L. Gillman and M. Jerison, Rings of continuous functions, University Series in Higher Math., Van Nostrand, Princeton, N. J., 1960. MR 22 #6994. MR 0116199 (22:6994)
  • [HH] H. Herrlich, E-kompakte Räume, Math. Z. 96 (1967), 228-255. MR 34 #5051. MR 0205218 (34:5051)
  • [HK] M. Hušek, The class of k-compact spaces is simple, Math. Z. 110 (1969), 123-126. MR 39 #6260. MR 0244947 (39:6260)
  • [HS1] D. Harris, Universal compact $ {T_1}$ spaces, General Topology and Appl. (to appear). MR 0331325 (48:9659)
  • [HS2] -, Transfinite metrics, sequences, and topological properties, Fund. Math. 73 (1971), 137-142. MR 0301709 (46:864)
  • [HS3] -, Structures in topology, Mem. Amer. Math. Soc. No. 115 (1971). MR 45 #5941.
  • [HS4] -, Closed images of the Wallman compactification, Proc. Amer. Math. Soc. (to appear). MR 0343238 (49:7982)
  • [HS5] -, The Wallman compactification as a functor, General Topology and Appl. 1 (1971), 273-281. MR 45 #1122. MR 0292034 (45:1122)
  • [HW] E. Hewitt, Rings of real-valued continuous functions. I, Trans. Amer. Math. Soc. 64 (1948), 45-99. MR 10, 126. MR 0026239 (10:126e)
  • [KE] E. Kamke, Theory of sets, Dover, New York, 1950. MR 0032709 (11:335a)
  • [NG] J. Nagata, Modern general topology, Bibliotheca Mathematica, vol. 7, North-Holland, Amsterdam; Wolters-Noordhoff, Groningen; Interscience, New York, 1968. MR 41 #9171. MR 0264579 (41:9171)
  • [SI1] R. Sikorski, On an ordered algebraic field, Comptes Rendue de la Societe des Science et des Lettres de Varsovie, Classe III, 1948, pp. 69-96. MR 0040274 (12:667f)
  • [SI2] -, Remarks on some topological spaces of high power, Fund. Math 37 (1950), 125-136. MR 12, 727. MR 0040643 (12:727e)
  • [SP] W. Sierpinski, Cardinal and ordinal numbers, 2nd rev. ed., Monografie Mat., vol. 34, PWN, Warsaw, 1965. MR 33 #2549. MR 0194339 (33:2549)
  • [WD] S. Willard, General topology, Addison-Wesley, Reading, Mass., 1970. MR 41 #9173. MR 0264581 (41:9173)
  • [ZS] O. Zariski and P. Samuel, Commutative algebra. Vols. I, II, University Series in Higher Math., Van Nostrand, Princeton, N. J., 1958, 1960. MR 19, 833; 22 #11006.

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DOI: https://doi.org/10.1090/S0002-9947-1974-0365492-4
Article copyright: © Copyright 1974 American Mathematical Society

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