Some remarks on the Bohr compactification of the number line (Q1088108)

The Bohr compactification B \({\mathbb{R}}\) of the reals \({\mathbb{R}}\) is under investigation. The crucial result of the paper is the theorem stating that B \({\mathbb{R}}\) is a completion of the uniform space \(({\mathbb{R}},U_{{\mathbb{R}}})\) with respect to the uniform structure \(U_{{\mathbb{R}}}\) generated by the family of pseudometrics \(\{r_{\lambda}\}_{\lambda \in {\mathbb{R}}\setminus \{0\}}\), where \(r_{\lambda}(x,y)=| e^{2\pi i\lambda x}-e^{2\pi i\lambda y}|\). However the topology on \({\mathbb{R}}\) generated by \(U_{{\mathbb{R}}}\) is strongly weaker than the usual topology of the reals. A uniformity \(U_ B\) on \({\mathbb{R}}\) is found such that \(U_ B\) generates the usual topology of the reals and the spaces B \({\mathbb{R}}\setminus {\mathbb{R}}\) and \(X\setminus {\mathbb{R}}\) are homeomorphic, where X is the completion of \(({\mathbb{R}},U_ B)\). Some results which describe the boundary of the Bohr compactification are given.