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Horrors of topology without AC: A nonnormal orderable space. (English) Zbl 0574.03039
Without AC there is a non-normal orderable space. More specifically, the statement ”\({\mathbb{Z}}\) many copies of \({\mathbb{Z}}\) fail to have a choice function” implies: ”the topological sum of \({\mathbb{Z}}\) many copies of \(\{- \infty \}\cup {\mathbb{Z}}\cup \{+\infty \}\) is not normal.” Note that this space is orderable. Further, the statement ”every orderable space is normal” is equivalent to a (possibly weaker) form of the axiom of choice, to wit: ”a pairwise disjoint family of non-empty convex open subsets of a complete linear order has a choice function.”
Reviewer: J.Roitman

03E30 Axiomatics of classical set theory and its fragments
54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
03E25 Axiom of choice and related propositions
06A05 Total orders
03E35 Consistency and independence results
06F30 Ordered topological structures (aspects of ordered structures)
54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)
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