Past exam of the probability, statistics and options I course of the University of Oxford / 2009 / E / 2 / b / i / Solution

The family contains the whole space and the empty set. A finite intersection of nontrivial members is $({\max }_j{a}_j,\infty )$. A nonempty union of rays is $({\inf }_j{a}_j,\infty )$ when the infimum is finite, and is the whole line when the infimum is $-\infty$. Empty unions and unions involving the whole space are already included. Hence $\tau$ is a topology. This is the upper-ray topology on the real line.