On Sequentially Compact Subspaces of ℝ without the Axiom of Choice

We show that the property of sequential compactness for subspaces of R is countably productive in ZF. Also, in the language of weak choice principles, we give a list of characterizations of the topological statement ‘sequentially compact subspaces of R are compact’. Furthermore, we show that forms 152 (= every non-well-orderable set is the union of a pairwise disjoint well-orderable family of denumerable sets) and 214 (= for every family A of infinite sets there is a function f such that for all y ∈ A, f (y) is a nonempty subset of y and | f (y)| = א0) of Howard and Rubin are equivalent.