We show that none of the following statements is provable in Zermelo-Fraenkel set theory (ZF) answering the corresponding open questions from Brunner in "The axiom of choice in topology": For every T 2 topological space (X, T ), if X is well-ordered, then each open cover of X has a well-ordered open refinement, (iii) For every T 2 topological space (X, T ), if X has a well-ordered dense subset, then there exists a function f : X × W → T such that W is a wellordered set and {x} = ∩ f ({x} × W ) for each x ∈ X.