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A Non-Hausdorff Multifunction Ascoli Theorem for 𝓴3-Spaces

Published online by Cambridge University Press:  20 November 2018

Pedro Morales
Pedro Morales*
Affiliation:
Université de Montréal, Montréal, Québec
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A non-Hausdorff Ascoli theorem for continuous functions was established in [6]. The present purpose is to extend this result to point-compact continuous multifunction, using Levine's generalization for closed subsets [12]. The paper is organized as follows: the object of section 2 is to establish the necessary multifunction lemmas and to introduce the notion of a Tychonoff set; section 3 generalizes to multifunction context the partial exponential law of R. H. Fox [9, p. 430], and establishes a special exponential law for multifunctions; section 4 concerns the crucial properties of even continuity for multifunctions, introduced in [8]; the main theorem of the paper is established in section 5.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 27 , Issue 4 , 01 August 1975 , pp. 893 - 900
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Arens, R., A topology for spaces of transformations, Ann. of Math. 7 (1946), 480–495.Google Scholar
2. Bagley, R. W. and Yang, J. S., On k-spaces and function spaces, Proc. Amer. Math. Soc. 17 (1966), 703–705.Google Scholar
3. Berge, C., Topological spaces (MacMillan Company, New York, 1965).Google Scholar
4. Brown, R., Function spaces and product topologies, Quart. J. Math. Oxford Ser. (2) 15 (1964), 238–250.Google Scholar
5. Cohen, D. E., Spaces with weak topology, Quart. J. Math. Oxford Ser. (2) 5 (1954), 77–80.Google Scholar
6. Fox, G. and Morales, P., A non-Hausdorff Ascoli theorem for C-spaces, Proc. Amer. Math. Soc. 39 (1973), 633–636.Google Scholar
7. Fox, G. and Morales, P., A general Tychonoff theorem for multifunctions, Can. Math. Bull. 17 (1974), to appear.Google Scholar
8. Fox, G. and Morales, P., Non-Hausdorff multifunction generalization of the Kelley-Morse Ascoli theorem (submitted to Pacific J. Math.).Google Scholar
9. Fox, R. H., On topologies for function spaces, Bull. Amer. Math. Soc. 51 (1945), 429–432.Google Scholar
10. Gale, D., Compact sets of functions and function rings, Proc. Amer. Math. Soc. 1 (1950), 303–308.Google Scholar
11. Kelley, J., General topology (D. van Nostrand, New York, 1965).Google Scholar
12. Levine, N., Generalized closed sets in topology, Rend. Circ. Mat. Palermo 19 (1970), 89–96.Google Scholar
13. Lin, Y. F., Tychonoff's theorem for the space of multif unctions, Amer. Math. Monthly 74 (1967), 399–400.Google Scholar
14. Lin, Y. F. and Rose, D. A., Ascoli1 s theorem for spaces of multif unctions, Pacific J. Math. 34 (1970), 741–747.Google Scholar
15. Mancuso, V. J., An Ascoli theorem for multi-valued functions, J. Austral. Math. Soc. 12 (1971), 466–472.Google Scholar
16. Morales, P., Pointwise compact spaces, Can. Math. Bull. 16 (1973), 545–549.Google Scholar
17. Myers, S. B., Equicontinuous sets of mappings, Ann. of Math. 47 (1946), 496–502.Google Scholar
18. Noble, N., Ascoli theorems and the exponential map, Trans. Amer. Math. Soc. 143 (1969), 393–411.Google Scholar
19. Noble, N., The continuity of functions on Cartesian products, Trans. Amer. Math. Soc. 149 (1970), 187–198.Google Scholar
20. Smithson, R. E., Topologies on sets of relations, J. Natur. Sci. and Math. (Lahore) 11 (1971), 43–50.Google Scholar
21. Smithson, R. E., Uniform convergence for multif unctions, Pacific J. Math. 39 (1971), 253–259.Google Scholar
22. Smithson, R. E., Multif unctions, Nieuw Arch. Wisk. 20 (1972). 31–53.Google Scholar