TITLE: Is there a non-metrizable topological space for which any countably compact subset is compact?
QUESTION [8 upvotes]: 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

REPLY [7 votes]: In addition to the above examples, it might be of interest that functional analysis is awash with such spaces.  For example, the space of distributions on a compact interval or that of functions analytic on a closed domain in the complex plane, both with their natural topologies.  More generally, any Silva space has this property.  A further ubiquitous source of examples is the dual of a separable Banach space with the bounded weak $\ast$ topology.  A suitable reference would be Köthe's classic on topological linear spaces.