On metric spaces with the properties of de Groot and Nagata in dimension one

TL;DR

This paper investigates metric spaces with de Groot and Nagata properties in one dimension, showing how these properties influence embeddings, space structure, and limitations on metric extensions.

Contribution

It proves that isometric embeddings of the interval into locally connected spaces with these properties are open, and explores implications for space structure and metric extension limitations.

Findings

Isometric embeddings of (0,1) are open in spaces with GP_1 or NP_1.

Euclidean metric cannot extend to an admissible GP_1-metric on the triode.

A GP_1-space containing all compact NP_1-spaces has continuum density.

Abstract

A metric space $(X,d)$ has the de Groot property $GP_n$ if for any points $x_0,x_1,...,x_{n+2}\in X$ there are positive indices $i,j,k\le n+2$ such that $i\ne j$ and $d(x_i,x_j)\le d(x_0,x_k)$. If, in addition, $k\in\{i,j\}$ then $X$ is said to have the Nagata property $NP_n$. It is known that a compact metrizable space $X$ has dimension $dim(X)\le n$ iff $X$ has an admissible $GP_n$-metric iff $X$ has an admissible $NP_n$-metric. We prove that an embedding $f:(0,1)\to X$ of the interval $(0,1)$ into a locally connected metric space $X$ with property $GP_1$ (resp. $NP_1$) is open provided $f$ is an isometric embedding (resp. $f$ has distortion $Dist(f)=\|f\|_\Lip\cdot\|f^{-1}\|_\Lip<2$). This implies that the Euclidean metric cannot be extended from the interval $[-1,1]$ to an admissible $GP_1$-metric on the triode $T=[-1,1]\cup[0,i]$. Another corollary says that a topologically homogeneous $GP_1$-space cannot contain an isometric copy of the interval $(0,1)$ and a topological copy of the triode $T$ simultaneously. Also we prove that a $GP_1$-metric space $X$ containing an isometric copy of each compact $NP_1$-metric space has density not less than continuum.