zbMATH — the first resource for mathematics

Countable choice and pseudometric spaces. (English) Zbl 0922.03068
The present paper is a survey about the set-theoretical strength (in the hierarchy of weak choice principles) of various topological assertions about pseudometric spaces. A typical example is Theorem 1.12: The countable choice axiom is equivalent to “Subspaces of separable pseudometric spaces are separable”.
Reviewer’s comment: A pseudometric is like a metric except that the distance of different points may vanish. Nevertheless, the authors’ results do not extend to metric spaces, where the strength of most assertions is not known; cf. P. Howard and J. E. Rubin: Consequences of the axiom of choice (AMS Surveys 59) (1998).
Reviewer: N.Brunner (Wien)

03E25 Axiom of choice and related propositions
54E35 Metric spaces, metrizability
54E52 Baire category, Baire spaces
Full Text: DOI
[1] Herrlich, H., Compactness and the axiom of choice, Appl. categ. struct., 4, 1-14, (1996) · Zbl 0881.54027
[2] H. Herrlich and J. Steprāns, Maximal filters, continuity, and choice principles, Quaestiones Math., to appear.
[3] H. Herrlich and G.E. Strecker, When is \(N\) Lindelöf?, Comment. Math. Univ. Carolin. 38, to appear.
[4] Jech, T., Eine bemerkung zum auswahlaxiom, Časopis Pěst. mat., 9, 30-31, (1968) · Zbl 0167.27402
[5] Jech, T., The axiom of choice, (1973), North-Holland Amsterdam · Zbl 0259.02051
[6] Moore, G.H., Zermelo’s axiom of choice—its origins, development and influence, (1982), Springer Berlin · Zbl 0497.01005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.