= Solution
The family contains the empty set and the entire <real line>. The <intersection> of two proper nonempty members is
$$
(a,\infty)\cap(b,\infty)=(\max(a,b),\infty),
$$
so 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.
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