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Where is pointwise multiplication in real $CK$-spaces locally open?

Volume 236 / 2017

Ehrhard Behrends Fundamenta Mathematicae 236 (2017), 51-69 MSC: 46B20, 46E15. DOI: 10.4064/fm121-1-2016 Published online: 2 August 2016

Abstract

Let $K$ be a compact Hausdorff space, and $B(f,r)$ the closed ball with center $f$ and radius $r$ in the space $CK$ of real-valued continuous functions on $K$.

Given $f,g\in CK$ we say that multiplication is locally open at $(f,g)$ if for every $\varepsilon \gt 0$ there exists a $\delta \gt 0$ such that $B(fg,\delta)\subset B(f,\varepsilon)B(g,\varepsilon)$; here $fg$ is the pointwise product of $f$ and $g$, and $B(f,\varepsilon)B(g,\varepsilon):=\{\tilde f \tilde g\mid \tilde f\in B(f,\varepsilon), \tilde g\in B(g,\varepsilon)\}$. For $K=[0,1]$ a characterization of the pairs $(f,g)$ at which multiplication is locally open is known. We extend it to arbitrary $K$.

Authors

  • Ehrhard BehrendsMathematisches Institut
    Freie Universität Berlin
    Arnimallee 6
    D-14195 Berlin, Germany
    e-mail

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