Previous |  Up |  Next

Article

Keywords:
Dugundji extension theorem; non-archimedean space; space of continuous functions; 0-dimensional space
Summary:
We prove a non-archimedean Dugundji extension theorem for the spaces $C^{\ast }(X,\mathbb {K})$ of continuous bounded functions on an ultranormal space $X$ with values in a non-archimedean non-trivially valued complete field $\mathbb {K}$. Assuming that $\mathbb {K}$ is discretely valued and $Y$ is a closed subspace of $X$ we show that there exists an isometric linear extender $T\colon C^{\ast }(Y,\mathbb {K})\rightarrow C^{\ast }(X,\mathbb {K})$ if $X$ is collectionwise normal or $Y$ is Lindelöf or $\mathbb {K}$ is separable. We provide also a self contained proof of the known fact that any metrizable compact subspace $Y$ of an ultraregular space $X$ is a retract of $X$.
References:
[1] Arens, R.: Extension of functions on fully normal spaces. Pac. J. Math. 2 (1952), 11-22. DOI 10.2140/pjm.1952.2.11 | MR 0049543 | Zbl 0046.11801
[2] Arkhangel'skij, A. V.: Spaces of mappings and rings of continuous functions. General topology III. Encycl. Math. Sci. 51 (1995), 71-156 Translation from Itogi Nauki tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 51 81-171 (1989). DOI 10.1007/978-3-662-07413-8_2 | MR 1036525 | Zbl 0826.54017
[3] Borges, C. J. R.: On stratifiable spaces. Pac. J. Math. 17 (1966), 1-16. DOI 10.2140/pjm.1966.17.1 | MR 0188982 | Zbl 0175.19802
[4] Groot, J. de: Non-archimedean metrics in topology. Proc. Am. Math. Soc. 7 (1956), 948-953. DOI 10.1090/S0002-9939-1956-0080905-8 | MR 0080905 | Zbl 0072.40201
[5] Dugundji, J.: An extension of Tietze's theorem. Pac. J. Math. 1 (1951), 353-367. DOI 10.2140/pjm.1951.1.353 | MR 0044116 | Zbl 0043.38105
[6] Ellis, R. L.: A non-Archimedean analogue of the Tietze-Urysohn extension theorem. Nederl. Akad. Wet., Proc., Ser. A 70 (1967), 332-333. MR 0212771 | Zbl 0148.16402
[7] Ellis, R. L.: Extending continuous functions on zero-dimensional spaces. Math. Ann. 186 (1970), 114-122. DOI 10.1007/BF01350686 | MR 0261565 | Zbl 0182.25501
[8] Engelking, R.: General Topology. Rev. and compl. ed. Sigma Series in Pure Mathematics, 6. Heldermann, Berlin (1989). MR 1039321 | Zbl 0684.54001
[9] Fabian, M., Habala, P., Hájek, P., Montesinos, V., Zizler, V.: Banach Space Theory. The Basis for Linear and Nonlinear Analysis. CMS Books in Mathematics. Springer, Berlin (2011). MR 2766381 | Zbl 1229.46001
[10] Gęba, K., Semadeni, Z.: Spaces of continuous functions. V: On linear isotonical embedding of $C(\Omega_1)$ into $C(\Omega_2)$. Stud. Math. 19 (1960), 303-320. MR 0117535 | Zbl 0094.30401
[11] Gruenhage, G., Hattori, Y., Ohta, H.: Dugundji extenders and retracts on generalized ordered spaces. Fundam. Math. 158 (1998), 147-164. MR 1656930 | Zbl 0919.54010
[12] Heath, R. W., Lutzer, D. J., Zenor, P. L.: On continuous extenders. Stud. Topol., Proc. Conf. Charlotte, N. C., 1974 203-213. MR 0358675 | Zbl 0312.54021
[13] Kuratowski, K.: Topology. Vol. II. Państwowe Wydawnictwo Naukowe Polish Scientific Publishers, Warsaw (1968). MR 0259835
[14] Michael, E.: Some extension theorem for continuous functions. Pac. J. Math. 3 (1953), 789-806. DOI 10.2140/pjm.1953.3.789 | MR 0059541
[15] Perez-Garcia, C., Schikhof, W. H.: Localy Convex Spaces Over Non-Archimedean Valued Fields. Cambridge University Press, Cambridge (2010). MR 2598517
[16] Douwen, E. K. van: Simultaneous linear extension of continuous functions. General Topology Appl. 5 (1975), 297-319. DOI 10.1016/0016-660X(75)90002-1 | MR 0380715
[17] Douwen, E. K. van, Lutzer, D. J., Przymusiński, T. C.: Some extensions of the Tietze-Urysohn theorem. Am. Math. Mon. 84 (1977), 435-441. DOI 10.2307/2321900 | MR 0458374
[18] Rooij, A. C. M. van: Non-Archimedean Functional Analysis. Monographs and Textbooks in Pure and Applied Mathematics. 51. Marcel Dekker, New York (1978). MR 0512894
Partner of
EuDML logo