Is there a sequential space that is compact but not sequentially compact? 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.

If $X$ is compact and sequential, it is in particular countably compact. And countable compactness and sequential compactness are equivalent for sequential spaces (I do believe we need at least $T_1$ here, but if your compact space is Hausdorff this is no problem). 
As noted in the comments, a sequential and sequentially compact Hausdorff space need not be compact (we only get to countable compactness; the Lindelöfness has to be assumed to get compactness), as shown by $\omega_1$. But from compactness we do get sequential compactness for sequential spaces.
