Show that the topologies of \(\mathbb{R}_\ell\) and \(\mathbb{R}_K\) are not comparable.
Solution
Recall that \(\mathbb R_\ell\) has basis \([a,b)\) and \(\mathbb R_K\) has basis consisting of ordinary intervals \((a,b)\) and sets \((a,b)-K\), where \(K=\{1/n:n\in\mathbb Z_+\}\).
First, \[ [0,1)\in\mathbb R_\ell \] but \([0,1)\) is not open in \(\mathbb R_K\): every \(\mathbb R_K\)-basic neighborhood of \(0\) contains negative points, so no such neighborhood is contained in \([0,1)\). Hence \(\mathbb R_\ell\) is not coarser than \(\mathbb R_K\).
Conversely, \[ (-1,1)-K \] is \(\mathbb R_K\)-open. It is not \(\mathbb R_\ell\)-open. Indeed \(0\) belongs to it, but every lower-limit basic neighborhood \([0,b)\) of \(0\) with \(b>0\) contains \(1/n\) for all sufficiently large \(n\), hence meets \(K\). Thus no lower-limit basic neighborhood of \(0\) lies inside \((-1,1)-K\). Therefore \(\mathbb R_K\) is not coarser than \(\mathbb R_\ell\) either.