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Among Hausdorff locally convex spaces, semi-reflexivity is characterized by the weak topology having the Heine-Borel property. It follows that, in a semi-reflexive space, every nested net of closed bounded nonempty convex sets has nonempty intersection.

Stated in terms of sequence, this is actually a characterization of reflexivity in the normed case, by a theorem of Smulian. Moreover, Klee extended the result from nonempty convex sets to star-shaped sets. My two questions are whether those results extend to Hausdorff locally convex spaces:

  1. If a Hausdorff locally convex space satisfies that every nested net of closed bounded nonempty convex sets has nonempty intersection, is it semi-reflexive?
  2. If a Hausdorff locally convex space satisfies the hypothesis of 1, does it actually satisfy that every nested net of closed bounded star-shaped sets has nonempty intersection?

For references, see:

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  • $\begingroup$ Semi-reflexivity is equivalent to the weak compactness of closed bounded subsets. $\endgroup$
    – Jochen Wengenroth
    Oct 8, 2021 at 5:09
  • $\begingroup$ Indeed... Corrected, thanks. $\endgroup$
    – Alcen
    Oct 8, 2021 at 14:24

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