Let me use the notation from "Axiom of Choice" by Horst Herrlich and refer to Theorem 4.55 from this book. I recall that CC(cR) is the statement that every nonempty countable family of complete nonempty subspaces of the real line R has a choice function. I cannot decide whether CC(cR) implies that, for each n>1, every complete subspace of R^n with its Euclidean metric is closed in R^n. Are the any already published results about this problem, obtained by other mathematicians ? I know that if CC(cR) fails, then there exists a simultaneously complete and connected subspace X of the plane R^2 such that X is not closed in R^2.
Regards, Eliza Wajch
I am very happy to inform you that Kyriakos Keremedis has proved the following theorem recently: It holds true in ZF that R^2 is sequential if and only if R is sequential. This immediately leads to positive answers to the questions posed in Problems 6.6 and 6.11 of the article "The minimizing vector theorem in symmetrized max-plus algebra" By C. Ozel. A. Piękosz, E. Wajch, H. Zekraoui (to appear in Jounal of Convex Analysis in 2019). I heartily congratulate K. Keremedis on his very nice theorem.
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