 Solution

The two conditions for a basis of a topology both hold. First, every real number $x$ belongs to $[x,x+1)$, so the proposed basic intervals cover the real line. Second, ifthen $\max (a,c)\le x<\min (b,d)$, andis itself one of the proposed basic intervals. Thus every point in an intersection of two basic sets has a basic neighbourhood contained in that intersection.

Consequently the arbitrary unions of members of $\mathcal{B}$, including the empty union, form a topology $\mathcal{T}$. The basis of a topology conditions give closure under finite intersections; arbitrary unions are built into the definition. This is the lower limit topology, also called the Sorgenfrey line.