Locally constant functions

Joan Hart; Kenneth Kunen

Fundamenta Mathematicae (1996)

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

Abstract

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Let X be a compact Hausdorff space and M a metric space. 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 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 as a normed linear space. We also build three first countable Eberlein compact spaces, F,G,H, with various properties. For all metric M, contains only the constant functions, and . If M is the Hilbert cube or any infinite-dimensional Banach space, then , but whenever for some finite n.

How to cite

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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

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