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

摘要

We 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).