Compact subsets of the upper-ray topology
= Compact subsets of the upper-ray topology
For the topology consisting of the empty set, the whole real line, and upper open rays, a nonempty subset is <compact> exactly when it has a smallest element. An open set containing the smallest element contains the entire subset. Conversely, if there is no smallest element, the rays $(a,\infty)$ indexed by points $a$ of the subset cover it, but any finite selection misses its smallest chosen endpoint.