Skip to main content Accessibility help
×
Home

Uniformities on a Product

  • Anthony W. Hager (a1)

Extract

All topological spaces shall be uniformizable (completely regular Hausdorff). A uniformity on X shall be viewed as a collection μ of coverings of X, via the manner of Tukey [20] and Isbell [16], and the associated uniform space denoted μX. Given the uniformizable topological space X, we shall be concerned with compatible uniformities as follows (discussed more carefully in § 1). The fine uniformity α (finest compatible with the topology); the “cardinal reflections“ αm of α (m an infinite cardinal number) ; αc, the weak uniformity generated by the real-valued continuous functions.

With μ standing, generically, for one of these uniformities, we consider the question: when is μ(X × Y) = μX × μY For μ = αℵ0 (the finest compatible precompact uniformity), the problem is equivalent to that of when

β(X × Y) = βX × βY,

β denoting Stone-Cech compactification; this is answered by the theorem of Glicksberg [9]. For μ = α, we have Isbell's generalization [16, VI1.32].

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Uniformities on a Product
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Uniformities on a Product
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Uniformities on a Product
      Available formats
      ×

Copyright

References

Hide All
1. Comfort, W. W., On the Hewitt realcompactification of a product space, Trans. Amer. Math. Soc. 131 (1968), 107118.
2. Comfort, W. W. and Anthony Hager, W., The projection and other continuous mappings on a product space, Math. Scand. 28 (1971), 7790.
3. Comfort, W. W. and Stelios Negrepontis, Extending continuous functions on X × Y to subsets of βX × βY, Fund. Math. 59 (1966), 112.
4. Corson, H. H. and Isbell, J. R., Euclidean covers of topological spaces, Quart. J. Math. Oxford Ser. 11 (1960), 3442.
5. Engelking, R., Outline of General Topology (North-Holland, Amsterdam, 1968),
6. Gantner, T. E., Extensions of uniformly continuous pseudometrics, Trans. Amer. Math. Soc. 132 (1968), 147157.
7. Gillman, L. and Jerison, M., Rings of continuous functions (Van Nostrand, Princeton, 1960).
8. Ginsberg, S. and Isbell, J. R., Some operators on uniform spaces, Trans. Amer. Math. Soc. 93 (1959), 145168.
9. Glicksberg, I., Stone-Čech compactifications of products, Trans. Amer. Math. Soc. 90 (1959), 369382.
10. Anthony W., Hager, Projections of zero-sets (and the fine uniformity on a product), Trans. Amer. Math. Soc. 140 (1969), 8794.
11. Anthony W., Hager, On inverse-closed subalgebras of C(X), Proc. London Math. Soc. 19 (1969), 233257.
12. Hewitt, E., Rings of real-valued continuous functions. I, Trans. Amer. Math. Soc.
13. Miroslav, Husek, The Hewitt realcompactification of a product, Comment. Math. Univ. Carolinae 11 (1970), 393395.
14. Miroslav, Husek, Pseudo-m-compactness and v[P × Q) (to appear in Indag. Math.).
15. Miroslav, Husek, Realcompactness of function spaces and v(P × Q) (to appear in Indag. Math.).
16. Isbell, J. R., Uniform spaces, Math. Surveys No. 12 (Amer. Math. Soc, Providence, 1964).
17. Noble, Norman, Products with closed projections, Trans. Amer. Math. Soc. 140 (1969).
18. Onuchic, Nelson, On the Nachbin uniform structure, Proc. Amer. Math. Soc. 11 (1960), 177-179.
19. Shirota, Taira, A class of topological spaces, Osaka Math. J. 1 (1952), 20-40.
20. Tukey, J. W., Convergence and uniformity in topology, Annals of Math. Studies, No. 2 (Princeton University Press, Princeton, 1940).
21. Vidossich, G., A note on cardinal reflections in the category of uniform spaces, Proc. Amer. Math. Soc. 23 (1969), 53-58.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed