On the minimal cover property in ZF

Abstract

We investigate in ZF set theory, i.e. Zermelo–Fraenkel set theory minus the Axiom of Choice AC, the set-theoretic strength of the following statementsMCC: Every topological space with the minimal cover property is compact and,MCP: For every infinite set X, the Tychonoff product2X, where2={0,1}is endowed with the discrete topology, has the minimal cover property.We also investigate the relationship between MCP, BPI (the Boolean prime ideal theorem), and Q(n) (for every infinite set X, the Tychonoff product 2X is n-compact), where n∈N, n≥2. We recall from [16] that BPI is equivalent to Q(n) for all integers n≥6.