Abstract

A metric space X\mathbf{X} is called densely complete if there exists a dense set DD in X\mathbf{X} such that every Cauchy sequence of points of DD converges in X\mathbf{X}. One of the main aims of this work is to prove that the countable axiom of choice, CAC\mathbf{CAC} for abbreviation, is equivalent to the following statements: (i) Every densely complete (connected) metric space X\mathbf{X} is complete. (ii) For every pair of metric spaces X\mathbf{X} and Y\mathbf{Y}, if Y\mathbf{Y} is complete and S\mathbf{S} is a dense subspace of X\mathbf{X}, while f ⁣:S→Yf\colon \mathbf{S}\rightarrow \mathbf{Y} is a uniformly continuous function, then there exists a uniformly continuous extension F ⁣:X→YF\colon \mathbf{X}\to \mathbf{Y} of ff. (iii) Complete subspaces of metric spaces have complete closures. (iv) Complete subspaces of metric spaces are closed. It is also shown that the restriction of (i) to subsets of the real line is equivalent to the restriction CAC(R)\mathbf{CAC}(\mathbb{R}) of CAC\mathbf{CAC} to subsets of R\mathbb{R}. However, the restriction of (ii) to subsets of R\mathbb{R} is strictly weaker than CAC(R)\mathbf{CAC}(\mathbb{R}) because it is equivalent to the statement that R\mathbb{R} is sequential. Moreover, among other relevant results, it is proved that, for every positive integer nn, the space Rn\mathbb{R}^n is sequential if and only if R\mathbb{R} is sequential. It is also shown that R×Q\mathbb{R}\times\mathbb{Q} is not densely complete if and only if CAC(R)\mathbf{CAC}(\mathbb{R}) holds.

Author information

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Kyriakos Keremedis

Dept. of Mathematics, University of the Aegean, Karlovassi, Samos 83200, Greece

kker@aegean.gr

Eliza Wajch

Institute of Mathematics, Faculty of Exact and Natural Sciences, Siedlce University of Natural Sciences and Humanities, 08-110 Siedlce, Poland

eliza.wajch@wp.pl

Keywords

  • Countable axiom of choice
  • complete metric spaces
  • connected metric spaces
  • sequential spaces

Mathematics Subject Classification

03E25, 54E35, 54E50, 54C20, 54D55

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Suggested citation

K. Keremedis, E. Wajch. “On Densely Complete Metric Spaces and Extensions of Uniformly Continuous Functions in ZF.” Journal of Convex Analysis 27 (2020), No. 4, 1099–1122. https://doi.org/10.68381/jca27058

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Published by Heldermann Verlag, 2020. Rights now held by Banach Press LLC.