Locally constant functions

Hart, Joan; Kunen, Kenneth

Fundamenta Mathematicae, Tome 149 (1996), p. 67-96 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $E_{0} (H, M) \neq C (H, M)$ , but $E_{0} (H, M) = C (H, M)$ whenever $M \subseteq ℝ^{n}$ for some finite n.

Publié le : 1996-01-01

Zbl 0855.46019

EUDML-ID : urn:eudml:doc:212164

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     author = {Joan Hart and Kenneth Kunen},
     title = {Locally constant functions},
     journal = {Fundamenta Mathematicae},
     volume = {149},
     year = {1996},
     pages = {67-96},
     zbl = {0855.46019},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv150i1p67bwm}
}

Hart, Joan; Kunen, Kenneth. Locally constant functions. Fundamenta Mathematicae, Tome 149 (1996) pp. 67-96. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv150i1p67bwm/

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