sequential compactness and countable compactness are equivalent properties. However, there exist examples of sequentially compact, first-countable spaces which are not compact (these are necessarily not metrizable spaces). One such space is the ordinal space Every first-countable space is compactly generated. Every subspace of a first-countable space is first-countable. Any countable product of a first-countable space is first-countable, although uncountable products need not be. See also References Bibliography General topology Properties of topological spaces