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The following is an ``old question in analysis:'' Is it true that every perfectly normal compact convex subset of a locally convex topological vector space is metrizable? Here perfectly normal means Hausdorff plus all closed subsets are a countable intersection of open sets.

Who first asked this question? The oldest reference I can locate is a 1972 paper by B. MacGibbon in the Journal of Functional Analysis but it is clear from what is written there that she is reporting progress on a known problem.

Of course I am also interested in an answer to this question, but I'm really asking about reference information. I should note that Lopez-Abad and Todorcevic have recently demonstrated that it is consistent with ZFC that there is a counterexample to this problem. The question is whether a positive answer is consistent.

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  • $\begingroup$ Just for the education of those less knowledgeable, could you provide a definition (or pointer) for a normal (compact convex) set? $\endgroup$
    – Joseph O'Rourke
    Commented Mar 12, 2011 at 2:34
  • $\begingroup$ @Joseph:"normal" as in separation property (aka T_4). Since this is about compact spaces, this is the same as Hausdorff (I edited normal->Hausdorff). $\endgroup$
    – Justin Moore
    Commented Mar 12, 2011 at 3:03

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