Normal functors in the category of non-Archimedean uniform spaces

A.Savchenko

We consider functors in the category of non-Archimedean uniform spaces and uniformly continuous maps generated by some normal functors in the category of compact Hausdorff spaces. We also show that any natural transformation of normal functors generates a natural transformation of the induced functors in the category of non-Archimedean uniform spaces.

1. J.W. de Bakker, J.I. Zucker, Processes and the denotational semantics of concurrency, Inform. and Control. 54 (1982) 70--120.

2. F. van Breugel, S.~Watson A Note on Hyperspaces and Terminal Coalgebras, CMCS'99, Coalgebraic Methods in Computer Science, Electronic Notes in Theoretical Computer Science Volume 19 (1999) 201--208.

3. B. S. Burdick, A note on completeness of hyperspaces // General topology and applications (Staten Island, NY, 1989), 19--24, Lecture Notes in Pure and Appl. Math., 134, Dekker, New York, 1991.

4. J.~Cao, H.P.~K\"unzi, I.L.~Reilly, S.~ Romaguera, Quasi-uniform hyperspaces of compact subsets, Topology Appl. 87 (1998) №2, 117--126.

5. A.~Chigogidze, Extension of normal functors, Vestnik Moskov. Univ. Ser. I Mat. Mekh. (1984) №6, 23--26 (in Russian).

6. V.V.~Fedorchuk, On completeness-type properties of spaces of $\tau$-additive probability measures, Vestnik Moskov. Univ. Ser. I Mat. Mekh. (1998) №5, 19--22 (in Russian).

7. J.I. den Hartog, E.P. de Vink, Building metric structures with the $Meas$ functor, preprint.

8. O.~Hubal', M.~Zarichnyi, Idempotent probability measures on ultrametric spaces, J. Math. Anal. Appl., 343 (2008) №2, 1052--1060.

9. J.R.~Isbell, Uniform spaces. Mathematical Surveys, No. 12 Amer. Math. Soc., Providence, R.I. 1964.

10. A.F.~Monna, Remarques sur les metriques non-archimediennes. II. (French) Nederl. Akad. Wetensch., Proc. 53, (1950) 625--637 (Indagationes Math. (1950) 179--191).

11. K. Morita, Completion of hyperspaces of compact subsets and topological completion of open-closed maps, Gen. Top. \& Appl., 4 (1974) 217--233.

12. I.V.~Protasov, Morphisms of ball's structures of groups and graphs, Ukrain. Mat. Zh. 54 (2002) №6, 847--855.

13. I. Protasov, M. Zarichnyi, General Asymptology, VNTL Publ., 2007.

14. Yu.V.~Sadovnichi, Uniform spaces of probability measures, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 50 (1995) №2, 85--87.

15. A. Savchenko, Normal functors in the category of ultrametric spaces, preprint, 2008.

16. E.V. Shchepin, Functors and uncountable powers of compacta, Uspekhi mat. nauk, 36 (1983) №3, 3--62.

17. A. van Rooij, Non-Archimedean Functional Analysis, Marcel Dekker, 1978.

18. A. C. M. van Rooij, Non-Archimedean uniformities. Kyungpook Math. J. 10 (1970) 21--30.

19. E.P. de Vink and J.J.M.M. Rutten. Bisimulation for probabilistic transition systems: a coalgebraic approach, Theoretical Computer Science, 221 (1999) 271--293.

20. A. J. Ward, Proof of a theorem of A. P. and W. Robertson on completeness under the Hausdorff uniformity. J. London Math. Soc. 18 (1978) №2, 349--352.

21. M.M.~Zarichnyi, Spaces of measures related to dequantization, J. Phys. Stud. 11 (2007) №1, 34--40.

	Normal functors in the category of non-Archimedean uniform spaces
Author	A.Savchenko Kherson State Agrarian University
Abstract	We consider functors in the category of non-Archimedean uniform spaces and uniformly continuous maps generated by some normal functors in the category of compact Hausdorff spaces. We also show that any natural transformation of normal functors generates a natural transformation of the induced functors in the category of non-Archimedean uniform spaces.
Keywords	normal functor, non-Archimedean uniform space, compact Hausdorff space
DOI	doi:10.30970/ms.31.2.165-171
Reference	1. J.W. de Bakker, J.I. Zucker, Processes and the denotational semantics of concurrency, Inform. and Control. 54 (1982) 70--120. 2. F. van Breugel, S.~Watson A Note on Hyperspaces and Terminal Coalgebras, CMCS'99, Coalgebraic Methods in Computer Science, Electronic Notes in Theoretical Computer Science Volume 19 (1999) 201--208. 3. B. S. Burdick, A note on completeness of hyperspaces // General topology and applications (Staten Island, NY, 1989), 19--24, Lecture Notes in Pure and Appl. Math., 134, Dekker, New York, 1991. 4. J.~Cao, H.P.~K\"unzi, I.L.~Reilly, S.~ Romaguera, Quasi-uniform hyperspaces of compact subsets, Topology Appl. 87 (1998) №2, 117--126. 5. A.~Chigogidze, Extension of normal functors, Vestnik Moskov. Univ. Ser. I Mat. Mekh. (1984) №6, 23--26 (in Russian). 6. V.V.~Fedorchuk, On completeness-type properties of spaces of $\tau$-additive probability measures, Vestnik Moskov. Univ. Ser. I Mat. Mekh. (1998) №5, 19--22 (in Russian). 7. J.I. den Hartog, E.P. de Vink, Building metric structures with the $Meas$ functor, preprint. 8. O.~Hubal', M.~Zarichnyi, Idempotent probability measures on ultrametric spaces, J. Math. Anal. Appl., 343 (2008) №2, 1052--1060. 9. J.R.~Isbell, Uniform spaces. Mathematical Surveys, No. 12 Amer. Math. Soc., Providence, R.I. 1964. 10. A.F.~Monna, Remarques sur les metriques non-archimediennes. II. (French) Nederl. Akad. Wetensch., Proc. 53, (1950) 625--637 (Indagationes Math. (1950) 179--191). 11. K. Morita, Completion of hyperspaces of compact subsets and topological completion of open-closed maps, Gen. Top. \& Appl., 4 (1974) 217--233. 12. I.V.~Protasov, Morphisms of ball's structures of groups and graphs, Ukrain. Mat. Zh. 54 (2002) №6, 847--855. 13. I. Protasov, M. Zarichnyi, General Asymptology, VNTL Publ., 2007. 14. Yu.V.~Sadovnichi, Uniform spaces of probability measures, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 50 (1995) №2, 85--87. 15. A. Savchenko, Normal functors in the category of ultrametric spaces, preprint, 2008. 16. E.V. Shchepin, Functors and uncountable powers of compacta, Uspekhi mat. nauk, 36 (1983) №3, 3--62. 17. A. van Rooij, Non-Archimedean Functional Analysis, Marcel Dekker, 1978. 18. A. C. M. van Rooij, Non-Archimedean uniformities. Kyungpook Math. J. 10 (1970) 21--30. 19. E.P. de Vink and J.J.M.M. Rutten. Bisimulation for probabilistic transition systems: a coalgebraic approach, Theoretical Computer Science, 221 (1999) 271--293. 20. A. J. Ward, Proof of a theorem of A. P. and W. Robertson on completeness under the Hausdorff uniformity. J. London Math. Soc. 18 (1978) №2, 349--352. 21. M.M.~Zarichnyi, Spaces of measures related to dequantization, J. Phys. Stud. 11 (2007) №1, 34--40.
Pages	165-171
Volume	31
Issue	2
Year	2009
Journal	Matematychni Studii
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