Abstract
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'':
(i) For every T2 topological space (X, T) if X is well-ordered, then X has a well-ordered base,
(ii) For every T2 topological space (X, T), if X is well-ordered, then there exists a function f : X × W $ \rightarrow$ T such that W is a well-ordered set and f ({x} × W) is a neighborhood base at x for each x $ \in$ X,
(iii) For every T2 topological space (X, T), if X has a well-ordered dense subset, then there exists a function f : X × W $ \rightarrow$ T such that W is a well-ordered set and {x} = $ \cap$ f ({x} × W) for each x $ \in$ X.
Citation
Kyriakos Keremedis. Eleftherios Tachtsis. "Nonconstructive Properties of Well-Ordered T2 topological Spaces." Notre Dame J. Formal Logic 40 (4) 548 - 553, Fall 1999. https://doi.org/10.1305/ndjfl/1012429718
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