The "World's Simplest Axiom of Choice" Fails.
Three Propositions Equivalent To The Axiom Of Choice
Tolerances, covering systems, and the axiom of choice
Two model theoretic ideas in independence proofs
Two theorems of functional analysis effectively equivalent to choice axioms
Tychonoff products of super second countable and super separable metric spaces
Tychonoff Products of Two-Element Sets and Some Weakenings of the Boolean Prime Ideal Theorem
Let X be an infinite set, and (X) the Boolean algebra of subsets of X. We consider the following statements: BPI(X): Every proper filter of (X) can be extended to an ultrafilter. UF(X): (X) has a free ultrafilter. We will show in ZF (i.e., Zermelo-Fraenkel set theory without the Axiom of Choice) that the following four statements are equivalent: (i) BPI(ω). (ii) The Tychonoff product , where 2 is the discrete space 0,1, is compact. (iii) The Tychonoff product is compact. (iv) In a Boolean algebra...
Tychonoff's theorem without the axiom of choice
Ultraproducts and the axiom of choice
Unique irredundant decompositions in upper continuous lattices
Universality of the μ-predictor
For suitable topological spaces X and Y, given a continuous function f:X → Y and a point x ∈ X, one can determine the value of f(x) from the values of f on a deleted neighborhood of x by taking the limit of f. If f is not required to be continuous, it is impossible to determine f(x) from this information (provided |Y| ≥ 2), but as the author and Alan Taylor showed in 2009, there is nevertheless a means of guessing f(x), called the μ-predictor, that will be correct except on a small set; specifically,...
Variants of the axiom of choice in set theory with atoms
When is Lindelöf?
Theorem. In ZF (i.e., Zermelo-Fraenkel set theory without the axiom of choice) the following conditions are equivalent: (1) is a Lindelöf space, (2) is a Lindelöf space, (3) is a Lindelöf space, (4) every topological space with a countable base is a Lindelöf space, (5) every subspace of is separable, (6) in , a point is in the closure of a set iff there exists a sequence in that converges to , (7) a function is continuous at a point iff is sequentially continuous at , (8)...
Words, free algebras, and coequalizers