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

Let ${\mathcal{B}}_{\mathrm{\ell }}=\{[a,b):a,b\in \mathbb{R},a<b\}$. The lower-limit topology ${\mathcal{T}}_{\mathrm{\ell }}$ on $\mathbb{R}$ is the topology having ${\mathcal{B}}_{\mathrm{\ell }}$ 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 ${\mathcal{B}}_{\mathrm{\ell }}$ determines a unique topology.

The lower-limit topology is finer than the usual topology: if $x\in (a,b)$, then $[x,(x+b)\mathrm{/}2)$ 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.