There are many topological results in Zermelo-Fraenkel set theory including the axiom of choice (ZFC) that are not true in the absence of choice, i. e. in ZF. Even if we restrict our attention to many “familiar” topological results are not provable in ZF, although in most cases their validity follows from a weaker version of the axiom of choice, CC( ). Definition 0.1 The axiom of countable choi...

The main theorem of this paper complements the tame-wild dichotomy for commutative Noetherian rings, obtained by Klingler and Levy [14]–[16]. They gave a complete classification of all finitely generated modules over Dedekind-like rings (cf. Definition 1.1) and showed that, over any ring that is not a homomorphic image of a Dedekind-like ring, the category of finite-length modules has wild repr...