Several results on compact metrizable spaces in $$\mathbf {ZF}$$

Abstract In the absence of axiom choice, set-theoretic status many natural statements about metrizable compact spaces is investigated. Some are provable in $$\mathbf {ZF}$$ ZF , some shown to be independent . For independence results, distinct models and permutation {ZFA}$$ ZFA with transfer theorems Pincus applied. New symmetric constructed each which power set $$\mathbb {R}$$ R well-orderable, Continuum Hypothesis satisfied but a denumerable family non-empty finite sets can fail have choice function, space need not embeddable into Tychonoff cube $$[0, 1]^{\mathbb {R}}$$ [ 0 , 1 ]