Locally constant functions
Fundamenta Mathematicae, Tome 150 (1996) no. 1, pp. 67-96
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.
@article{10_4064_fm_150_1_67_96,
author = {Joan Hart and Kenneth Kunen},
title = {Locally constant functions},
journal = {Fundamenta Mathematicae},
pages = {67--96},
year = {1996},
volume = {150},
number = {1},
doi = {10.4064/fm-150-1-67-96},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/fm-150-1-67-96/}
}
Joan Hart; Kenneth Kunen. Locally constant functions. Fundamenta Mathematicae, Tome 150 (1996) no. 1, pp. 67-96. doi: 10.4064/fm-150-1-67-96
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