Some independence results about compact metrizable spaces and two notions of finiteness

Abstract

We settle some open problems from Keremedis, Tachtsis and Wajch (2021) [16] and from Keremedis and Tachtsis (2001) [15].In particular, we establish the following results:1.There is a permutation model of ZFA in which every linearly ordered set can be well ordered, the axiom of choice for denumerable families of non-empty finite sets and the axiom of countable multiple choice are false, and there exists a compact, dense-in-itself, non-second countable, weakly separable, metrizable space (X,τ) with a base for τ that can be written as a union of a denumerable family of finite sets, but the family of all non-empty closed subsets of X has no multiple choice function in the model.2.There is a model of ZF which has the same properties as the model of (1), except for “every linearly ordered set can be well ordered” which is false in the model.3.There is a model of ZF in which there is a filter base infinite set X which is strongly filter base finite, and thus ω+1 with the order topology is not a remainder of the discrete space (X,P(X)) in the model. For the independence results (2) and (3), we will use certain permutation models of ZFA and then we will apply the Jech–Sochor First Embedding Theorem in order to transfer those results into ZF. Specifically, the model of (1) (recently constructed by Howard and Tachtsis ((2021) [8])) will be used for (2), while for (3) we will introduce a new permutation model and prove that it satisfies the proposition given in (3).