Notre Dame Journal of Formal Logic

Nonconstructive Properties of Well-Ordered T2 topological Spaces

Kyriakos Keremedis and Eleftherios Tachtsis


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.

Article information

Notre Dame J. Formal Logic, Volume 40, Number 4 (1999), 548-553.

First available in Project Euclid: 30 January 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03E25: Axiom of choice and related propositions
Secondary: 03E35: Consistency and independence results 54Gxx: Peculiar spaces


Keremedis, Kyriakos; Tachtsis, Eleftherios. Nonconstructive Properties of Well-Ordered T 2 topological Spaces. Notre Dame J. Formal Logic 40 (1999), no. 4, 548--553. doi:10.1305/ndjfl/1012429718.

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