Nonconstructive Properties of Well-Ordered T2 topological Spaces

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.