Separable connected metric spaces need not have continuum size in ZF

摘要

We show: (ⅰ) It is relatively consistent with ZF that there exists a connected, separable subspace C of R~2 with |C| ＜ |R|. (ⅱ) It is relatively consistent with ZF that there exists a separable, connected, compact, pseudometric (X,d) with |X| ＜ |R|. (ⅲ) It is relatively consistent with ZF that there exists a separable, compact, connected, pseudometric space (X, d) whose size is strictly less than the power of its metric reflection (X~*,d~*). (ⅳ) It is relatively consistent with ZF that there exists a connected, locally connected, non-pathwise connected, compact, non-separable pseudometric space (A,d) such that A is Dedekind-finite and its metric reflection is the interval [0, 1]. (ⅴ) It is relatively consistent with ZF~0 (= ZF minus the axiom of regularity) that there exists a non-separable, compact, connected, locally connected metric space. (ⅵ) Every subspace X of Hilbert's cube [0, 1]~N such that XX is meager in X is separable. In particular, every connected subspace X of [0,1]~N such that XX is meager in X is separable. (ⅶ) Every connected subspace X of [0,1]~N such that XX is meager in X has continuum size. (ⅷ) The countable axiom of choice restricted to subsets of the real line R, CAC(R) is equivalent to the proposition: "Every connected subspace X of R~2 is separable". (ⅸ) Every family A = (A_i)_(i∈R) of non-empty sets has a choice set iff every connected, locally connected, compact pseudometric space is pathwise connected.