MAXIMAL FILTERS, CONTINUITY AND CHOICE PRINCIPLES

Abstract

In Zermelo-Fraenkel set theory ZF without the Axiom of Choice AC, results such as the following are established: 1. The Boolean Prime Ideal Theorem BPI is equivalent to the statement: ZFE Every zero-filter is contained in a maximal one. 2. The Boolean Prime Ideal Theorem is properly weaker than the statement: CFE Every closed filter is contained in a maximal one. 3. The Axiom of Choice is equivalent to the conjunction of CFE and the Axiom of Countable Choice CC. 4. The Axiom of Countable Choice is equivalent to the statement: C=SC Functions between metric spaces are continuous iff they are seqentially continuous.