# What are the sets on which norm-closedness implies weakly closedness?

Let $$X$$ be a Banach space. Let $$C$$ be a convex, and normed-closed subset of $$X$$. It is well-known that $$C$$ becomes weakly closed subset of $$X$$. I want to know is there any well-know class of non convex sets which has this property?

i.e., a class of sets in $$X$$, not necessary convex on which, norm-closed implies weakly-closed.

• Totally bounded sets have this property. – Sergei Akbarov Jun 20 at 6:20
• @SergeiAkbarov Yeah.. but that's a trivial case , because in banach space closed totally bounded sets are compact. – Red shoes Jun 20 at 6:41
• The proof for the convex case is by the Hahn-Banach theorem. For a new, substantially different, case: you will probably need a new, substantially different, proof. – Gerald Edgar Jun 20 at 9:48
• Another trivial case: the union of finitely many closed convex sets. – Robert Israel Jun 20 at 15:53
• @GeraldEdgar What if set be norm-locally convex? – Red shoes Jun 20 at 20:23