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1996 | 150 | 1 | 67-96
Tytuł artykułu

Locally constant functions

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EN
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Let X be a compact Hausdorff space and M a metric space. $E_0(X,M)$ is the set of f ∈ C(X,M) such that there is a dense set of points x ∈ X with f constant on some neighborhood of x. We describe some general classes of X for which $E_0(X,M)$ is all of C(X,M). These include βℕ\ℕ, any nowhere separable LOTS, and any X such that forcing with the open subsets of X does not add reals. In the case where M is a Banach space, we discuss the properties of $E_0(X,M)$ as a normed linear space. We also build three first countable Eberlein compact spaces, F,G,H, with various $E_0$ properties. For all metric M, $E_0(F,M)$ contains only the constant functions, and $E_0(G,M) = C(G,M)$. If M is the Hilbert cube or any infinite-dimensional Banach space, then $E_0(H,M) ≠ C(H,M)$, but $E_0(H,M) = C(H,M)$ whenever $M ⊆ ℝ^n$ for some finite n.
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autor
  • Department of Mathematics, University of Wisconsin Madison, Wisconsin 53706 U.S.A., jhart@math.wisc.edu
  • Department of Mathematics, University of Wisconsin Madison, Wisconsin 53706 U.S.A., kunen@cs.wisc.edu
Bibliografia
  • [1] A. V. Arkhangel'skiĭ and V. V. Fedorchuk, General Topology I, Basic Concepts and Constructions, Dimension Theory, Springer, 1990.
  • [2] A. Bella, A. Hager, J. Martinez, S. Woodward and H. Zhou, Specker spaces and their absolutes, I, preprint.
  • [3] A. Bella, J. Martinez and S. Woodward, Algebras and spaces of dense constancies, preprint.
  • [4] Y. Benyamini, M. E. Rudin and M. Wage, Continuous images of weakly compact subsets of Banach spaces, Pacific J. Math. 70 (1977), 309-324.
  • [5] A. Bernard, Une fonction non Lipschitzienne peut-elle opérer sur un espace de Banach de fonctions non trivial?, J. Funct. Anal. 122 (1994), 451-477.
  • [6] A. Bernard, A strong superdensity property for some subspaces of C(X), prépublication de l'Institut Fourier, Laboratoire de Mathématiques, 1994.
  • [7] A. Bernard and S. J. Sidney, Banach like normed linear spaces, preprint, 1994.
  • [8] M. Džamonja and K. Kunen, Properties of the class of measure separable compact spaces, Fund. Math. 147 (1995), 261-277.
  • [9] P. R. Halmos, Lectures on Boolean Algebras, Van Nostrand, 1963.
  • [10] T. Jech, Set Theory, Academic Press, 1978.
  • [11] K. Kunen, Set Theory, North-Holland, 1980.
  • [12] J. Martinez and S. Woodward, Specker spaces and their absolutes, II, Algebra Universalis, to appear.
  • [13] J. van Mill, A homogeneous Eberlein compact space which is not metrizable, Pacific J. Math. 101 (1982), 141-146.
  • [14] M. E. Rudin and W. Rudin, Continuous functions that are locally constant on dense sets, J. Funct. Anal. 133 (1995), 120-137.
  • [15] S. J. Sidney, Some very dense subspaces of C(X), preprint, 1994.
  • [6] R. Sikorski, Boolean Algebras, Springer, 1964.
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Bibliografia
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bwmeta1.element.bwnjournal-article-fmv150i1p67bwm
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