In second-countable topological spaces, the three notions of compactness, countable compactness, and sequential compactness all coincide. This follows from the Lindel"{o}f property: a second-countable space is Lindel"{o}f, so every open cover has a countable subcover, which means countable compactness implies compactness. Combined with the fact that sequential compactness always implies countable compactness, all three notions become equivalent.