Is this kind of space metrizable? It has a nice result from Tkachuk V V. Spaces that are projective with respect to classes of mappings[J]. Trans. Moscow Math. Soc, 1988, 50: 139-156.

If the closure of every discrete subset of a space is compact then the whole space is compact.

The proof can be seen here by Brian M. Scott.
Then these questions arise naturally:
Question 1: If the closure of every discrete subset of a space is countably compact, then  is the whole space countably compact?
Question 2: If the closure of every discrete subset of a space is metrizable, then is the whole space metrizable?

Q1: (Assuming $T_1$-ness) If $X$ were not countably compact then there is a countable subset $A$ without a limit point. But this $A$ is then closed and discrete, so $A$ is the closure of a discrete set, hence countably compact, which it obviously is not. So the question seems trivial for the countably compact case. 
Q2: There is a consistent counterexample at least (I studied that space in the past) in this paper (not free) by Kenneth Kunen. But I think there should be a much easier example.
