If \(L\) is a straight line in the plane, describe the topology \(L\) inherits as a subspace of \(\mathbb{R}_\ell \times \mathbb{R}\) and as a subspace of \(\mathbb{R}_\ell \times \mathbb{R}_\ell\). In each case it is a familiar topology.
Solution
Parametrize a nonvertical line \(L=\{(x,mx+k):x\in\mathbb R\}\) by its first coordinate \(x\), and a vertical line by its second coordinate.
For \(\mathbb R_\ell\times\mathbb R\), a basic rectangle is \[ [a,b)\times(c,d). \] If \(L\) is vertical, intersection with such rectangles gives ordinary open intervals in the second coordinate, so the induced topology is the standard topology on \(\mathbb R\). If \(L\) is nonvertical (including horizontal lines), the trace of \([a,b)\times(c,d)\) on \(L\) is \[ \{x\in[a,b):mx+k\in(c,d)\}, \] the intersection of \([a,b)\) with a standard open set, hence open in \(\mathbb R_\ell\). Conversely, \([x_0,x_0+\varepsilon)\) is the trace of \([x_0,x_0+\varepsilon)\times(c,d)\) for any open interval \((c,d)\) containing \(\{mx+k:x_0\le x\le x_0+\varepsilon\}\). So the induced topology is the lower-limit topology. Hence \[ L\cong \begin{cases} \mathbb R,&L\text{ vertical},\\ \mathbb R_\ell,&L\text{ nonvertical}. \end{cases} \]
For \(\mathbb R_\ell\times\mathbb R_\ell\), basic rectangles are \[ [a,b)\times[c,d). \] A vertical or horizontal line inherits \(\mathbb R_\ell\). A line of positive slope \(m\) also inherits \(\mathbb R_\ell\): the trace of \([a,b)\times[c,d)\) is \[ [a,b)\cap\Bigl[\tfrac{c-k}{m},\tfrac{d-k}{m}\Bigr), \] a half-open interval, and \([x_0,x_0+\varepsilon)\) is the trace of \([x_0,x_0+\varepsilon)\times[mx_0+k,\,mx_0+k+m\varepsilon)\). A line of negative slope is discrete. Indeed, at \((x_0,y_0)\) on such a line, for sufficiently small \(\varepsilon,\delta>0\) the rectangle \[ [x_0,x_0+\varepsilon)\times[y_0,y_0+\delta) \] meets the line only at \((x_0,y_0)\), since moving to larger \(x\) forces \(y<y_0\). Therefore \[ L\cong \begin{cases} \mathbb R_\ell,&L\text{ vertical, horizontal, or of positive slope},\\ \mathbb R_d,&L\text{ of negative slope}. \end{cases} \]