Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-28T08:33:27.655Z Has data issue: false hasContentIssue false
  • English
  • Français

A Hahn-Banach Theorem in Subbase Convexity Theory

Published online by Cambridge University Press:  20 November 2018

M. van de Vel
M. van de Vel*
Affiliation:
Vrije Universiteit, Amsterdam, Holland
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In the last fifteen years, topology has shown up with an increasing interest in the use of closed subbases. Starting from Frink's internal characterization of complete regularity (Frink [6]), DeGroot and Aarts used closed subbases to obtain Hausdorff compactifications of completely regular spaces, thus giving a characterization of the latter in terms of their subbases [1]. The main tool of that paper is the notion of a linked system, which naturally leads to the notions of supercompactness and superextensions [7]. After 1970, these two topics developed to indepedennt theories, with several deep results available at this moment. Most results up to 1976 are summarized in [12].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 4 , April 1980 , pp. 804 - 820
Copyright
Copyright © Canadian Mathematical Society 1980

References

1. Aarts, J. M. and de Groot, J., Complete regularity as a separation axion, Can. J. Math. 21 (1969), 96–105.Google Scholar
2. Bell, M. G. and van Mill, J., The compactness number of a compact topological space, (to appear in Fund. Math.).CrossRefGoogle Scholar
3. Curtis, D. W. and Schori, R. M., 2X and C(X) are homeomorphic to the Hilbert cube, Bull. Amer. Math. Soc. 80 (1974), 927–931.Google Scholar
4. van Douwen, E. K., Special bases for compact metrizable spaces, (to appear in Fund. Math.).CrossRefGoogle Scholar
5. van Douwen, E. K. and van Mill, J., Supercompact spaces, (to appear in Gen. Top. Appl.).CrossRefGoogle Scholar
6. Frink, O., Compactifications and semi-normal spaces, Amer. J. Math. 86 (1964), 602–607.Google Scholar
7. de Groot, J., Supercompactness and superextensions, Contributions to extension theory of topological structures, Symp. Berlin 1967 (Deutsche Verlag de Wissenschaften, Berlin, 1969), 89–90.Google Scholar
8. Grothendieck, A., Topological vector spaces (Gordon and Breach, N.Y., 1973).Google Scholar
9. Kay, D. C. and Womble, E. W., Axiomatic convexity theory and relationships between the Caratheodory, Kelly, and Radon Number, Pacific J. Math. 38 (1971), 471–485.Google Scholar
10. van Mill, J., The superextension of the closed unit interval is homeomorphic to the Hilbert cube, Fund. Math. 103 (1978), 151–175.Google Scholar
11. van Mill, J., Superextensions of metrizable continua are Hilbert cubes, (to appear in Fund. Math.).CrossRefGoogle Scholar
12. van Mill, J., Supercompactness and \Voilman spaces, M. C. tract 85 (Mathematisch Centrum, Amsterdam, 1977).Google Scholar
13. van Mill, J. and Schrijver, A., Subbase characterizations of compact topological spaces, Gen. Top. Appl. 10 (1979), 183–201.Google Scholar
14. van Mill, J. and van de Vel, M., Subbases, convex sets, and hyperspaces, (to appear in Pacific J. Math.)CrossRefGoogle Scholar
15. van Mill, J. and van de Vel, M., Convexity preserving mappings in subbase convexity theory, Proc. Kon. Ned. Acad. Wet. 81 (1978), 76–90.Google Scholar
16. van Mill, J. and van de Vel, M., On superextensions and hyperspaces, Topological Structures II, Math. Centre tract 115, Amsterdam (1979), 769–180.Google Scholar
17. van Mill, J. and Wattel, E., An external characterization of spaces which admit normal binary subbases, Amer. J. Math. 100 (1978), 987–994.Google Scholar
18. Mills, C. F., A simpler proof that compact metric spaces are supercompact, Proc. Amer. Math. Soc. 73 (1979), 388–390.Google Scholar
19. Schori, R. M. and West, I. E., 2I is homeomorphic to the Hilbert cube, Bull. Am. Math. Soc. 78 (1972), 402–406.Google Scholar
20. Strok, M. and Szymansky, A., Compact metric spaces have binary bases, Fund. Math. 89 (1975), 81–91.Google Scholar
21. van de Vel, M., Superextensions and Lefschetz fixed point structures, Fund. Math. 104 (1978), 33–48.Google Scholar
22. Verbeek, A., Superextensions of topological spaces, M.C. tract 41, Amsterdam.Google Scholar