Topology Proceedings 33 (2009), pp. 239-249

On super second countable and super separable metric spaces

Horst Herrlich, Kyriakos Keremedis and Eleftherios Tachtsis

In the framework of ZF (i.e., Zermelo-Fraenkel set theory without the Axiom of Choice) we show that: (1) Every super second countable metric space is super separable. (2) Every super second countable metric space is hereditarily super second countable. The above results answer related questions from Gutierres "On first and second countable spaces and the axiom of choice". We also show that the axiom CAC(R) (i.e., the Axiom of Choice restricted to countable families of non-empty subsets of reals) is equivalent to the converse of (1) and to the corresponding statement of (2) for super separable metric spaces.

Keywords: axiom of choice, weak axioms of choice, second countable metric spaces, super second countable metric spaces, separable metric spaces, super separable metric spaces

Mathematics Subject Classification: 03E25 54A35 (54D65 54D70 54E35)

Document formats: PDF file 177.8 Kb

Topology Proceedings, Volume 33 (2009)
Subscription information