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Hewitt Realcompactifications of Products

Published online by Cambridge University Press:  20 November 2018

William G. McArthur
William G. McArthur*
Affiliation:
The Pennsylvania State University, University Park, Pennsylvania Shippensburg State Collège, Shippensburg, Pennsylvania
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The Hewitt realcompactification vX of a completely regular Hausdorff space X has been widely investigated since its introduction by Hewitt [17]. An important open question in the theory concerns when the equality v(X × Y) = vX × vY is valid. Glicksberg [10] settled the analogous question in the parallel theory of Stone-Čech compactifications: for infinite spaces X and Y, β(X × Y) = βX × β Y if and only if the product X × Y is pseudocompact. Work of others, notably Comfort [3; 4] and Hager [13], makes it seem likely that Glicksberg's theorem has no equally specific analogue for v(X × Y) = vX × vY. In the absence of such a general result, particular instances may tend to be attacked by ad hoc techniques resulting in much duplication of effort.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 3 , 01 June 1970 , pp. 645 - 656
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Anderson, F. W. and Blair, R. L., Characterizations of the algebra of all real-valued continuous functions on a completely regular space, Illinois J. Math. 3 (1959), 121–133.Google Scholar
2. Arens, R. F. and Kaplansky, I., Topological representation of algebras, Trans. Amer. Math. Soc. 63 (1948), 457–481.Google Scholar
3. Comfort, W. W., Locally compact realcompactifications, pp. 95-100, General Topology and Its Relations to Modern Analysis and Algebra. II, Proceedings of the Second Prague Topological Symposium, 1966 (Academic Press, New York, 1967).Google Scholar
4. Comfort, W. W., On the Hewitt realcompactification of a product space, Trans. Amer. Math. Soc. 131 (1968), 107–118.Google Scholar
5. Comfort, W. W. and Negrepontis, S., Extending continuous functions on X X Y to subsets of /3X X |8F, Fund. Math. 59 (1966), 1–12.Google Scholar
6. Engelking, R., Remarks on realcompact spaces, Fund. Math. 55 (1964), 303–308.Google Scholar
7. Frolik, Z., The topological product of two pseudocompact spaces, Czech. Math. J. 10 (1960), 339–349.Google Scholar
8. Gillman, L. and Jerison, M., Stone-Cech compactification of a product, Arch. Math. 10 (1959), 443–446.Google Scholar
9. Gillman, L. and Jerison, M., Rings of continuous functions, The University Series in Higher Mathematics (Van Nostrand, Princeton, N.J., 1960).Google Scholar
10. Glicksberg, I., Stone-£ech compactification of products, Trans. Amer. Math. Soc. 90 (1958), 369–382.Google Scholar
11. Hager, A. W., On the tensor product of function rings, Doctoral Dissertation, Pennsylvania State University, University Park, 1965.Google Scholar
12. Hager, A. W., Some remarks on the tensor product of function rings, Math. Z. 92 (1966), 210–224.Google Scholar
13. Hager, A. W., On inverse-closed subalgebras of C(X) (to appear).Google Scholar
14. Hager, A. W., Projections of zero-sets (and thefine uniformity on a product), Trans. Amer. Math. Soc. 140 (1969), 87–94.Google Scholar
15. Hager, A. W. and Mrowka, S. G., Compactness and the projection mapping from a product space, Notices Amer. Math. Soc. 12 (1965), 368.Google Scholar
16. Henriksen, M., On the equivalence of the ring, lattice, and semigroup of continuous functionst Proc. Amer. Math. Soc. 7 (1956), 959–960.Google Scholar
17. Hewitt, E., Rings of real-valued continuous functions, I, Trans. Amer. Math. Soc. 64 (1948), 45–99.Google Scholar
18. Isbell, J. R., Uniform Spaces (Amer. Math. Soc, Providence, R.I., 1964).Google Scholar
19. Kelley, J. L., General topology (Van Nostrand, Princeton, N.J.-Toronto-New York- London, 1955).Google Scholar
20. Onuchic, N., On the Nachbin uniform structure, Proc. Amer. Math. Soc. 11 (1960), 177–179.Google Scholar
21. Tamano, H., A note on the pseudo-compactness of the product of two spaces, Mem. Col. Sci. Univ. Kyoto Ser. A Math. 83 (1960/61), 225–230.Google Scholar