A basis for $\mathbb{R}_{l}\times\mathbb{R}$ is the collection of sets $[a,b)\times(c,d)$ . A basis for $\mathbb{R}_{l}\times\mathbb{R}_{l}$ is the collection of sets $[a,b)\times[c,d)$ . In both cases the intersection of a basis set and the line is an interval on the line, either $[)$ or $()$ or $(]$ or $[]$ (we consider the line as a copy of $\mathbb{R}$ having a particular direction in the plane). And vice versa, that is if any such interval can be obtained as the intersection of the line with some basis set, then all same type intervals on the line can be obtained as the intersection of the line with some basis sets.

Now, not all combinations of these intervals are possible in all cases. If the line is vertical, then the respective intervals are $()$ (produced by the intersection with $[a,b)\times(c,d)$ ), and $[)$ or $(]$ (produced by the intersection with $[a,b)\times(c,d)$ , and depending on the direction of the line), and respective topologies are $\mathbb{R}$ , and $\mathbb{R}_{l}$ or $\mathbb{R}_{u}$ . If the line is horizontal, then in both cases the topology is either $\mathbb{R}_{l}$ or $\mathbb{R}_{u}$ depending on the direction of the line. If it has some slope then the first topology may generate either ($[)$ and $()$ ) or ($(]$ and $()$ ), depending on the direction of the line. That is, it is either the lower limit or the upper limit topology. For the second topology, if the line is downward sloping, we have intersections of the form $[]$ , $[)$ , $(]$ and $()$ , i.e. the discrete topology. Finally, if the line is upward sloping then the second topology generates a basis consisting of either $[)$ or $(]$ , i.e. the topology is either $\mathbb{R}_{l}$ or $\mathbb{R}_{u}$ .

Direction	$\mathbb{R}_{l}\times\mathbb{R}$	$\mathbb{R}_{l}\times\mathbb{R}_{l}$
$\uparrow$	$\mathbb{R}$	$\mathbb{R}_{l}$
$\nearrow$	$\mathbb{R}_{l}$	$\mathbb{R}_{l}$
$\rightarrow$	$\mathbb{R}_{l}$	$\mathbb{R}_{l}$
$\searrow$	$\mathbb{R}_{l}$	$\mathbb{R}_{d}$
$\downarrow$	$\mathbb{R}$	$\mathbb{R}_{u}$
$\swarrow$	$\mathbb{R}_{u}$	$\mathbb{R}_{u}$
$\leftarrow$	$\mathbb{R}_{u}$	$\mathbb{R}_{u}$
$\nwarrow$	$\mathbb{R}_{u}$	$\mathbb{R}_{d}$