# Locally constant functions

Fundamenta Mathematicae (1996)

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

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topHart, 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 -

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