Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
2009 | 7 | 4 | 670-682
Tytuł artykułu

A Hilbert cube compactification of the function space with the compact-open topology

Treść / Zawartość
Warianty tytułu
Języki publikacji
Let X be an infinite, locally connected, locally compact separable metrizable space. The space C(X) of real-valued continuous functions defined on X with the compact-open topology is a separable Fréchet space, so it is homeomorphic to the psuedo-interior s = (−1, 1)ℕ of the Hilbert cube Q = [−1, 1]ℕ. In this paper, generalizing the Sakai-Uehara’s result to the non-compact case, we construct a natural compactification $$ \bar C $$(X) of C(X) such that the pair ($$ \bar C $$(X), C(X)) is homeomorphic to (Q, s). In case X has no isolated points, this compactification $$ \bar C $$(X) coincides with the space USCCF(X,) of all upper semi-continuous set-valued functions φ: X → = [−∞, ∞] such that each φ(x) is a closed interval, where the topology for USCCF(X, ) is inherited from the Fell hyperspace Cld*F(X × ) of all closed sets in X × .
  • Graduate School of Mathematical Sciences, University of Tsukuba, Tsukuba, 305-8571, Japan
  • [1] Beer G., Topologies on closed and closed convex sets, Kluwer Acad. Publ., Dordrecht, 1993
  • [2] Beer G., On the Fell topology, Set-Valued Anal., 1993, 1, 69–80[Crossref]
  • [3] Bessaga C., Pełczyński A., Selected topics in infinite-dimensional topology, MM 58, Polish Sci. Publ., Warsaw, 1975
  • [4] Bing R.H., Partioning a set, Bull. Amer. Math. Soc., 1949, 55, 1101–1110[Crossref]
  • [5] Chapman T.A., Dense sigma-compact subsets of infinite-dimensional manifolds, Trans. Amer. Math. Soc., 1971, 154, 399–426[Crossref]
  • [6] Engelking R., General topology (Revised and complete edition), Sigma Ser. in Pure Math., Heldermann Verlag, Berlin, 1989
  • [7] Fedorchuk V.V., On certain topological properties of completions of function spaces with respect to Hausdorff uniformity, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1991, 77–80 (in Russian); English translation: Moscow Univ. Math. Bull., 1991, 46, 56–58
  • [8] Fedorchuk V.V., Completions of spaces of functions on compact spaces with respect to the Hausdorff uniformity, Tr. Semin. im. I. G. Petrovskogo, 1995, 18, 213–235 (in Russian); English translation: J. of Math. Sci., 1996, 80, 2118–2129
  • [9] Fell J.M.G., A Hausdorff topology for the closed subsets of a locally compact non-Hausdorff space, Proc. Amer. Math. Soc., 1962, 13, 472–476[Crossref]
  • [10] Moise E.E., Grille decomposition and convexification theorems for compact locally connected continua, Bull. Amer. Math. Soc., 1949, 55, 1111–1121[Crossref]
  • [11] van Mill J., Infinite-dimensional topology, Elsevier Sci. Publ. B.V., Amsterdam, 1989
  • [12] Sakai K., Uehara S., A Hilbert cube compactification of the Banach space of continuous functions, Topology Appl., 1999, 92, 107–118[Crossref]
  • [13] Sakai K., Yang Z., Hyperspaces of non-compact metrizable spaces which are homeomorphic to the Hilbert cube, Topology Appl., 2002, 127, 331–342[Crossref]
  • [14] Tominaga A., One some properties of non-compact Peano spaces, J. Sci. Hiroshima Univ., Ser.A, 1956, 19(3), 457–467
  • [15] Tominaga A., Tanaka T., Convexification of locally connected generalized continua, J. Sci. Hiroshima Univ., Ser.A, 1955, 19(2), 301–306
Typ dokumentu
Identyfikator YADDA
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.