We show that the property of sequential compactness for subspaces of 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 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.
"On Sequentially Compact Subspaces of without the Axiom of Choice." Notre Dame J. Formal Logic 44 (3) 175 - 184, 2003. https://doi.org/10.1305/ndjfl/1091030855