A characterization of Valdivia compact spaces

• Autores: Ondrej F. K. Kalenda
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 51, Fasc. 1, 2000, págs. 59-82
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • We characterize Valdivia compact spaces $K$ in terms of $\mathcal{C}(K)$ endowed with a topology introduced by M. Valdivia (1991). This generalizes R. Pol’s characterization of Corson compact spaces. Further we study duality, products and open continuous images of Valdivia compact spaces. We prove in particular that the dual unit ball of $\mathcal{C}(K)$ is Valdivia whenever $K$ is Valdivia and that the converse holds whenever $K$ has a dense set of $G_\delta$ points. Another result is that any open continuous image of a Valdivia compact space with a dense set of $G_\delta$ points is again Valdivia.

• Referencias bibliográficas
  • A.V. Arhangel’skiĭ, Topological spaces of functions, Moscow State Univ., 1989 (in Russian); Kluwer Acad. Publ. Dordrecht-Boston-London, 1992...
  • S. Argyros, S. Mercourakis and S. Negrepontis, Functional analytic properties of Corson compact spaces, Studia Math. 89 (1988), 197–229.
  • R. Deville and G. Godefroy, Some applications of projective resolutions of identity, Proc. London Math. Soc. (3) 67(1) (1993), 183–199.
  • R. Deville, G. Godefroy and V. Zizler, Smoothness and renormings in Banach spaces, Pitman Monographs 64, 1993.
  • N. Dunford and J.T. Schwartz, Linear operators, Part I, Interscience publishers, New York, London, 1958.
  • M. Fabian, Gâteaux differentiability of convex functions and topology: weak Asplund spaces, Wiley-Interscience, New York, 1997.
  • M. Fabian, G. Godefroy and V. Zizler, A note on Asplund generated spaces, Bull. Polish. Acad. Sci. 47(2) (1999), to appear.
  • I. Glicksberg, Stone-Cech compactifications of products, Trans. Amer. Math. Soc. 90 (1959), 369–382.
  • O. Kalenda, An example concerning Valdivia compact spaces, Serdica Math. J. 25 (1999), 131–140.
  • O. Kalenda, Continuous images and other topological properties of Valdivia compacta, Fund. Math., to appear.
  • A.S. Kechris, Classical Descriptive Set Theory, Springer-Verlag, 1995.
  • N. Noble, The continuity of functions on Cartesian products, Trans. Amer. Math. Soc. 149 (1970), 187–198.
  • B.E. Shapirovskii, Special types of embeddings in Tychonoff cubes, Subspaces of $\Sigma$-products and cardinal invariants, Collection: Topology,...
  • A. Valdivia, Projective resolutions of the identity in $C(K)$ spaces, Arch. Math. (Basel) 54 (1990), 493–498.
  • M. Valdivia, Simultaneous resolutions of the identity operator in normed spaces, Collect. Math. 42(3) (1991), 265–285.
  • M. Valdivia, On certain compact topological spaces, Rev. Mat. Univ. Complut. Madrid 10(1), (1997), 81–84.