Separable connected metric spaces need not have continuum size in ZF

We show: (i) It is relatively consistent with ZF that there exists a connected, separable subspace C of R 2 with | C | < | R | . t is relatively consistent with ZF that there exists a separable, connected, compact, pseudometric ( X , d ) with | X | < | R | . 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 ⁎ ) . t 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 ] . is relatively consistent with ZF 0 (=ZF minus the axiom of regularity) that there exists a non-separable, compact, connected, locally connected metric space. very subspace X of Hilbertʼs cube [ 0 , 1 ] N such that X ¯ \ X is meager in X ¯ is separable. In particular, every connected subspace X of [ 0 , 1 ] N such that X ¯ \ X is meager in X ¯ is separable. Every connected subspace X of [ 0 , 1 ] N such that X ¯ \ X is meager in X ¯ has continuum size. 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”. very 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.