Find all the compact sets of the lower limit topology in $\mathbb{R}$, i.e. the topology given by basis $\{[a,b)|a,b \in \mathbb{R} \}$.
What I have got so far:
necessary conditions: the compact sets of lower limit topology must be countable, also they are closed (in the usual sense) and bounded. Also all of the limit points must be approached above, i.e. for every limit point $L$, there exists $\epsilon$ such that there are no points within $(L-\epsilon,L)$
sufficient condition: countable sets with finitely many limit points, each of the limit points approached above (as defined above) are compact sets in lower limit topology; countable sets with countably many limit points, where the limit points themselves have finitely many limit points, are also compact sets in lower limit topology; as well as countable sets with countably many limit points, the limit points having countably many limit points, the limit points of limit points having finitely many limit points, etc…
As you can see, I am quite close to finding a necessary and sufficient condition, if only I can prove it is impossible to have sets which have countably many limit points, the limit points having countably many limit points,… (descending indefinitely)
Any help is appreciated!
Best Answer
Thanks @Brian M. Scott for the reference, I have figured out the question, for other's reference, I'll state it here:
S is compact w.r.t. lower limit topology iff S is compact w.r.t. usual topology and S is well ordered with respect to $\geq$ (i.e. every subset of S has a largest element) iff S is compact w.r.t. usual topology, S is countable and every limit point is approached above