On the minimal cover property in ZF

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 statements very topological space with the minimal cover property is compact and, or every infinite set X, the Tychonoff product 2 X , where 2 = { 0 , 1 } is endowed with the discrete topology, has the minimal cover property. o investigate the relationship between MCP, BPI (the Boolean prime ideal theorem), and Q ( n ) (for every infinite set X, the Tychonoff product 2 X is n-compact), where n ∈ N , n ≥ 2 . We recall from [16] that BPI is equivalent to Q ( n ) for all integers n ≥ 6 .