 Solution

The family contains the empty set and the entire real line. The intersection of two proper nonempty members isso it is closed under finite intersection. For an arbitrary nonempty family of proper rays, put $c={\inf }_i{a}_i$. If the thresholds are unbounded below, the union is $\mathbb{R}$; otherwise the union is $(c,\infty )$. In the latter case $c$ itself is in no ray, and every point greater than $c$ lies above some threshold, proving the equality. Empty unions and unions involving $\mathbb{R}$ cause no difficulty. Thus arbitrary unions also belong to the family, establishing the topology axioms.