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

Ehrhard Behrends¹

¹ Mathematisches Institut Freie Universität Berlin Arnimallee 6 D-14195 Berlin, Germany

Fundamenta Mathematicae, Tome 236 (2017) no. 1, pp. 51-69

Résumé

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 $K$.

Détail

DOI : 10.4064/fm121-1-2016

Keywords: compact hausdorff space closed ball center radius space real valued continuous functions given say multiplication locally every varepsilon there exists delta delta subset varepsilon varepsilon here pointwise product varepsilon varepsilon tilde tilde mid tilde varepsilon tilde varepsilon characterization pairs which multiplication locally known extend arbitrary