 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.