On the relative strength of forms of compactness of metric spaces and their countable productivity in ZF

AbstractWe show in ZF that:(i)A countably compact metric space need not be limit point compact or totally bounded and, a limit point compact metric space need not be totally bounded.(ii)A complete, totally bounded metric space need not be limit point compact or Cantor complete.(iii)A Cantor complete, totally bounded metric space need not be limit point compact.(iv)A second countable, limit point compact metric space need not be totally bounded or Cantor complete.(v)A sequentially compact, selective metric space (the family of all non-empty open subsets of the space has a choice function) is compact.(vi)A countable product of sequentially compact (resp. compete and totally bounded) metric spaces is sequentially compact (resp. compete and totally bounded)