Notre Dame Journal of Formal Logic 40 (4):548-553 (1999)

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 T such that W is a well-ordered set and f ({x} × W) is a neighborhood base at x for each x 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 T such that W is a well-ordered set and {x} = f ({x} × W) for each x X
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DOI 10.1305/ndjfl/1012429718
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References found in this work BETA

[Omnibus Review].Kenneth Kunen - 1969 - Journal of Symbolic Logic 34 (3):515-516.
The Axiom of Choice in Topology.Norbert Brunner - 1983 - Notre Dame Journal of Formal Logic 24 (3):305-317.
[Omnibus Review].Thomas Jech - 1992 - Journal of Symbolic Logic 57 (1):261-262.

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