The title is the question : Is there a non-metrizable topological space for which any countably compact subset is compact ? 


EDIT : **non-metrizable and Hausdorff**



Is there a non-metrizable topological space for which any countably compact subset is compact?
Any Lindelöf non-metrizable Hausdorff space will do (EDIT: you need that the space is C-closed as well, see below), but more generally, a space is called isocompact iff every closed countably compact subset of X is compact, cl-isocompact iff the closure of a countably compact subset is compact, and C-closed iff any countably compact subset is closed. This paper by Cho and Park contains various results on (cl)-isocompactess, as well a other references:


<http://www.mathnet.or.kr/mathnet/kms_tex/978637.pdf>


And this one by Ismail and Nyikos has good informations on C-closed spaces:


<http://www.sciencedirect.com/science/article/pii/0166864180900279>


