k-Spaces, sequential spaces and related topics in the absence of the axiom of choice

In the absence of the axiom of choice, among other new results concerning sequential, Fréchet-Urysohn, Loeb, Cantor completely metrizable, k-spaces and very k-spaces, it is proved that every Loeb T3-space having a base expressible as a countable union of finite sets is a metrizable second-countable space all of whose Fσ-subspaces are separable; every Gδ-subspace of a second-countable, Cantor completely metrizable space is Cantor completely metrizable, Loeb and separable. Arkhangel'skii's statement that every very k -space is Fréchet-Urysohn is unprovable in ZF but it holds in ZF that every first-countable, regular very k-space whose family of all non-empty compact sets has a choice function is Fréchet-Urysohn. That every second-countable metrizable space is a very k-space is equivalent to the axiom of countable choice for R.