Semirings and $T_{1}$ compactifications. I
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- by Douglas Harris PDF
- Trans. Amer. Math. Soc. 188 (1974), 241-258 Request permission
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.References
-
N. Bourbaki, General topology. Part I, Addison-Wesley, Reading, Mass., 1966. MR 34 #5044b.
- N. J. Fine, L. Gillman, and J. Lambek, Rings of quotients of rings of functions, McGill University Press, Montreal, Que., 1966. MR 0200747
- Leonard Gillman and Meyer Jerison, Rings of continuous functions, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0116199, DOI 10.1007/978-1-4615-7819-2
- Horst Herrlich, ${\mathfrak {E}}$-kompakte Räume, Math. Z. 96 (1967), 228–255 (German). MR 205218, DOI 10.1007/BF01124082
- Miroslav Hušek, The class of $k$-compact spaces is simple, Math. Z. 110 (1969), 123–126. MR 244947, DOI 10.1007/BF01124977
- Douglas Harris, Universal compact $T_{1}$ spaces, General Topology and Appl. 3 (1973), 291–318. MR 331325, DOI 10.1016/0016-660X(73)90018-4
- Douglas Harris, Transfinite metrics, sequences and topological properties, Fund. Math. 73 (1971/72), no. 2, 137–142. MR 301709, DOI 10.4064/fm-73-2-137-142 —, Structures in topology, Mem. Amer. Math. Soc. No. 115 (1971). MR 45 #5941.
- Douglas Harris, Closed images of the Wallman compactification, Proc. Amer. Math. Soc. 42 (1974), 312–319. MR 343238, DOI 10.1090/S0002-9939-1974-0343238-9
- Douglas Harris, The Wallman compactification as a functor, General Topology and Appl. 1 (1971), 273–281. MR 292034, DOI 10.1016/0016-660X(71)90098-5
- Edwin Hewitt, Rings of real-valued continuous functions. I, Trans. Amer. Math. Soc. 64 (1948), 45–99. MR 26239, DOI 10.1090/S0002-9947-1948-0026239-9
- E. Kamke, Theory of Sets. Translated by Frederick Bagemihl, Dover Publications, Inc., New York, N.Y., 1950. MR 0032709
- Jun-iti Nagata, Modern general topology, Bibliotheca Mathematica, Vol. VII, North-Holland Publishing Co., Amsterdam; Wolters-Noordhoff Publishing, Groningen; Interscience Publishers John Wiley & Sons, Inc., New York, 1968. MR 0264579
- Roman Sikorski, On an ordered algebraic field, Soc. Sci. Lett. Varsovie. C. R. Cl. III. Sci. Math. Phys. 41 (1948), 69–96 (1950) (English, with Polish summary). MR 40274
- Roman Sikorski, Remarks on some topological spaces of high power, Fund. Math. 37 (1950), 125–136. MR 40643, DOI 10.4064/fm-37-1-125-136
- Wacław Sierpiński, Cardinal and ordinal numbers, Second revised edition, Monografie Matematyczne, Vol. 34, Państwowe Wydawnictwo Naukowe (PWN), Warsaw, 1965. MR 0194339
- Stephen Willard, General topology, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1970. MR 0264581 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.
Additional Information
- © Copyright 1974 American Mathematical Society
- 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