Compact subsets of a cocountable space
= Compact subsets of a cocountable space
Only finite subsets are compact in the <cocountable topology>. An infinite subset contains distinct $c_1,c_2,\ldots$. The open sets obtained by removing the tails $\{c_j:j\geq n\}$ form an increasing <open cover>, but every finite selection misses a tail point. Finite sets are compact in every topology.