Locally constant functions

Joan Hart; Kenneth Kunen

Fundamenta Mathematicae (1996)

  • Volume: 150, Issue: 1, page 67-96
  • ISSN: 0016-2736

Abstract

top
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, 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.

How to cite

top

Hart, Joan, and Kunen, Kenneth. "Locally constant functions." Fundamenta Mathematicae 150.1 (1996): 67-96. <http://eudml.org/doc/212164>.

@article{Hart1996,
abstract = {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, 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.},
author = {Hart, Joan, Kunen, Kenneth},
journal = {Fundamenta Mathematicae},
keywords = {locally constant functions; totally ordered set; nowhere separable LOTS; order topology; Eberlein compact spaces; Hilbert cube},
language = {eng},
number = {1},
pages = {67-96},
title = {Locally constant functions},
url = {http://eudml.org/doc/212164},
volume = {150},
year = {1996},
}

TY - JOUR
AU - Hart, Joan
AU - Kunen, Kenneth
TI - Locally constant functions
JO - Fundamenta Mathematicae
PY - 1996
VL - 150
IS - 1
SP - 67
EP - 96
AB - 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, 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.
LA - eng
KW - locally constant functions; totally ordered set; nowhere separable LOTS; order topology; Eberlein compact spaces; Hilbert cube
UR - http://eudml.org/doc/212164
ER -

References

top
  1. [1] A. V. Arkhangel'skiĭ and V. V. Fedorchuk, General Topology I, Basic Concepts and Constructions, Dimension Theory, Springer, 1990. 
  2. [2] A. Bella, A. Hager, J. Martinez, S. Woodward and H. Zhou, Specker spaces and their absolutes, I, preprint. Zbl0856.54039
  3. [3] A. Bella, J. Martinez and S. Woodward, Algebras and spaces of dense constancies, preprint. Zbl1079.54506
  4. [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. Zbl0374.46011
  5. [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. [6] A. Bernard, A strong superdensity property for some subspaces of C(X), prépublication de l'Institut Fourier, Laboratoire de Mathématiques, 1994. 
  7. [7] A. Bernard and S. J. Sidney, Banach like normed linear spaces, preprint, 1994. 
  8. [8] M. Džamonja and K. Kunen, Properties of the class of measure separable compact spaces, Fund. Math. 147 (1995), 261-277. Zbl1068.28502
  9. [9] P. R. Halmos, Lectures on Boolean Algebras, Van Nostrand, 1963. 
  10. [10] T. Jech, Set Theory, Academic Press, 1978. 
  11. [11] K. Kunen, Set Theory, North-Holland, 1980. 
  12. [12] J. Martinez and S. Woodward, Specker spaces and their absolutes, II, Algebra Universalis, to appear. Zbl0886.54031
  13. [13] J. van Mill, A homogeneous Eberlein compact space which is not metrizable, Pacific J. Math. 101 (1982), 141-146. Zbl0495.54020
  14. [14] M. E. Rudin and W. Rudin, Continuous functions that are locally constant on dense sets, J. Funct. Anal. 133 (1995), 120-137. 
  15. [15] S. J. Sidney, Some very dense subspaces of C(X), preprint, 1994. 
  16. [6] R. Sikorski, Boolean Algebras, Springer, 1964. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.