The lower-limit topology on $\mathbb{R}$, with the half-open intervals $[a,b)$ as a basis

Let $\mathcal B_\ell=\{[a,b):a,b\in\mathbb R,\ a<b\}$. The lower-limit topology $\mathcal T_\ell$ on $\mathbb R$ is the topology having $\mathcal B_\ell$ as a basis. The resulting space is the lower-limit line.

This basis is well defined. It covers $\mathbb R$, because $x\in[x,x+1)$ for every $x$. If $x\in[a,b)\cap[c,d)$, then $x\in[\max(a,c),\min(b,d))$, whose right endpoint exceeds $x$ and which lies inside the intersection. Thus the two basis conditions of A family is a basis for a unique topology iff it covers the set and every point of an intersection of two members lies in a member inside that intersection; finite intersections of any subbasis form a basis hold, so $\mathcal B_\ell$ determines a unique topology.

The lower-limit topology is finer than the usual topology: if $x\in(a,b)$, then $[x,(x+b)/2)$ is a lower-limit basic interval containing $x$ and contained in $(a,b)$. No equality with the usual topology is asserted here. The half-open intervals use the interval convention of Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length, and opens are exactly unions of basis members by Basis and subbasis for a topology, and the topology generated by a family of sets.