$\mathbf{E}$-compact extensions in the absence of the Axiom of Choice

Abstract

The main aim of this work is to show, in the absence of the Axiom of Choice, fundamental results on $\mathbf{E}$-compact extensions of $\mathbf{E}$-completely regular spaces, in particular, on Hewitt realcompactifications and Banaschewski compactifications. Some original results concern a special subring of the ring of all continuous real functions on a given zero-dimensional $T_1$-space. New facts about $P$-spaces, Baire topologies and $G_{\delta }$-topologies are also shown. Not all statements investigated here have proofs in $\mathbf{ZF}$. Some statements are shown equivalent to the Boolean Prime ideal Theorem, some are consequences of the Axiom of Countable Multiple Choices.