I am trying to understand if there is an error in this question, or if the answer is trivial.
Define a topology on $[0,\infty)$ with open sets
$(a,\infty), a \in (0,\infty)$
$[0,\infty)$ and $\emptyset$.
Show $[0,\infty)$ is compact in this topology.
Since we can only build an open cover out of open sets, surely we can only take $\cup_{n > 0} (n,\infty)$ which does not cover $[0,\infty)$.
Can we either:
Trivially take $[0,\infty)$ as an open finite cover, meaning whenever we take the whole space of any topology it is automatically compact.
The question should read $[a,\infty), a \in [0,\infty)$ are open in this topology
Any help would be greatly appreciated.
Many thanks,
Ash