MATH 582, INTRODUCTION TO SET THEORY, WINTER 1999 Problem Set 9 — Zorn’s Lemma

Problem 1 If ⟨P, ⩽⟩ is a partially ordered set (=poset), then a chain is a subset C ⊆ P such that for all x, y ∈ C, x ⩽ y or y ⩽ x. Show that any poset has a maximal chain, i.e., a chain that is not a proper subset of any other chain. Problem 2 Prove Zorn’s lemma for posets: If ⟨P, ⩽⟩ is a poset such that any chain has an upper bound in P, then P has a maximal element. Hint: use Problem 1. Problem 3 Give a direct proof of the Well-Ordering Theorem from Zorn’s Lemma. Hint: Consider the set

𝒜 = {R ⊆ A × A | R is a well-order on some subset of A.}

Problem 4 A subset of the Euclidean plane, A ⊆ ℝ × ℝ, is called convex if for any two points u, v ∈ A, the line segment connecting u and v is contained in A. In other words, A ⊆ ℝ × ℝ is convex iff for all u, v ∈ A and for all λ ∈ ℝ with 0 ⩽ λ ⩽ 1,

λu + (1 − λ)v ∈ A.

(a) Prove that any subset X of the Euclidean plane has a maximal convex subset.

(b) Give at least three different examples of maximal convex subsets of the following subset of ℝ × ℝ: X = {⟨x, y⟩ | (|x| ⩽ 1 ∧ |y| ⩽ 3) ∨ (|x| ⩽ 3 ∧ |y| ⩽ 1)}.

Problem 5 (Teichmüller-Tukey lemma) Assume that 𝒜 is a nonempty set such that for every set B,

B ∈ 𝒜 ⟺ every finite subset of B is a member of 𝒜.

Show that 𝒜 has a maximal element, i.e., an element that is not a subset of any other element of 𝒜.