Partition models, Permutations of infinite sets without fixed points, and weak forms of AC

Banerjee, Amitayu (2023) Partition models, Permutations of infinite sets without fixed points, and weak forms of AC. COMMENTATIONES MATHEMATICAE UNIVERSITATIS CAROLINAE. ISSN 0010-2628

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Abstract

Abstract. In set theory without the Axiom of Choice (AC), we observe new relations of the following statements with weak choice forms. • There does not exist an infinite Hausdorff space X such that every infinite subset of X contains an infinite compact subset. • If a field has an algebraic closure then it is unique up to isomorphism. • For every set X there is a set A such that there exists a choice function on the collection [A] 2 of two-element subsets of A and satisfying |X| ≤ |2 [A] 2 |. • Van Douwen’s Choice Principle (Every family X = {(Xi, ≤i) : i ∈ I} of linearly ordered sets isomorphic with (Z, ≤) has a choice function, where ≤ is the usual ordering on Z). We also extend the research works of B.B. Bruce [4]. Moreover, we prove that the principle “Any infinite locally finite connected graph has a spanning m-bush for any even integer m ≥ 4” is equivalent to K˝onig’s Lemma in ZF (i.e., the Zermelo–Fraenkel set theory without AC). We also give a new combinatorial proof to show that any infinite locally finite connected graph has a chromatic number if and only if K˝onig’s Lemma holds.

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