A space $X$ is
- sequential if every sequentially closed set is closed,
- compact, if every open cover has a finite subcover,
- sequentially compact, if every sequence has a convergent subsequence.
Any first-countable space is sequential. Any first-countable compact space is sequentially compact.
The typical example $I^I$ with $I = [0,1]$ for a compact not sequentially compact space is not sequential. For such a counterexample it is necessary that $X$ is not first-countable.
The database version of Steen-Seebach $\pi$-base does not provide such an example.