For what kind of topological vector spaces (separable maybe?) are the closed convex subsets *countable* intersections of halfspaces. 


I've seen somewhere that it's true for separable Hilbert spaces, but without proof or reference. Is there a reference on this fact (and related questions)? 



When are the closed convex subsets countable intersections of halfspaces
This question is addressed in:


*Borwein, Jonathan M.; Vanderwerff, Jon D.*, [**Constructible convex sets**](http://dx.doi.org/10.1023/B:SVAN.0000023393.75251.05), Set-Valued Anal. 12, No. 1-2, 61-77 (2004). [ZBL1059.46056](https://zbmath.org/?q=an:1059.46056).


