A+ CATEGORY SCIENTIFIC UNIT

PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

Uniform continuity and normality of metric spaces in $\mathbf{ZF}$

Volume 65 / 2017

Kyriakos Keremedis Bulletin Polish Acad. Sci. Math. 65 (2017), 113-124 MSC: 03E25, 54E35, 54E45, 54E50, 54C20, 54C30. DOI: 10.4064/ba8122-10-2017 Published online: 23 October 2017

Abstract

Let $\mathbf{X}=(X,d)$ and $\mathbf{Y}=(Y,\rho )$ be two metric spaces.

(a) We show in $\mathbf{ZF}$ that:

(i) If $\mathbf{X}$ is separable and $f:\mathbf{X}\rightarrow \mathbf{Y}$ is a continuous function then $f$ is uniformly continuous iff for any $ A,B\subseteq X$ with $d(A,B)=0$, $\rho (f(A),f(B))=0$. But it is relatively consistent with $\mathbf{ZF}$ that there exist metric spaces $\mathbf{X}$, $ \mathbf{Y}$ and a continuous, non-uniformly continuous function $f:\mathbf{X} \rightarrow \mathbf{Y}$ such that for any $A,B\subseteq X$ with $d(A,B)=0$, $\rho (f(A),f(B))=0$.

(ii) If $S$ is a dense subset of $\mathbf{X}$, $\mathbf{Y}$ is Cantor complete and $f:\mathbf{S}\rightarrow \mathbf{Y}$ a uniformly continuous function, then there is a unique uniformly continuous function $F:\mathbf{X} \rightarrow \mathbf{Y}$ extending $f$. But it is relatively consistent with $\mathbf{ZF}$ that there exist a metric space $\mathbf{X}$, a complete metric space $\mathbf{Y}$, a dense subset $S$ of $\mathbf{X}$ and a uniformly continuous function $f:\mathbf{S} \rightarrow \mathbf{Y}$ that does not extend to a uniformly continuous function on $\mathbf{X}$.

(iii) $\mathbf{X}$ is complete iff for any Cauchy sequences $ (x_{n})_{n\in \mathbb{N}}$ and $(y_{n})_{n\in \mathbb{N}}$ in $\mathbf{X}$, if $\overline{ \{x_{n}:n\in \mathbb{N}\}}\cap \overline{\{y_{n}:n\in \mathbb{N}\}} =\emptyset $ then $d(\{x_{n}:n\in \mathbb{N}\},\{y_{n}:n\in \mathbb{N} \}) \gt 0 $.

(b) We show in $\mathbf{ZF}$+$\mathbf{CAC}$ that if $f:\mathbf{X} \rightarrow \mathbf{Y}$ is a continuous function, then $f$ is uniformly continuous iff for any $A,B\subseteq X$ with $d(A,B)=0$, $\rho (f(A),f(B))=0$.

Authors

  • Kyriakos KeremedisDepartment of Mathematics
    University of the Aegean
    Karlovassi, Samos 83200, Greece
    e-mail

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image