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

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.