# On metric spaces with the properties of de Groot and Nagata in dimension one - Mathematics > Metric Geometry

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$i e j$ and $dx i,x j\le dx 0,x k$. If, in addition, $k\in\{i,j\}$ then$X$ is said to have the Nagata property $NP n$. It is known that a compactmetrizable space $X$ has dimension $dimX\le n$ 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 alocally connected metric space $X$ with property $GP 1$ resp. $NP 1$ is openprovided $f$ is an isometric embedding resp. $f$ has distortion$Distf=\|f\| \Lip\cdot\|f^{-1}\| \Lip<2$. This implies that the Euclideanmetric cannot be extended from the interval $-1,1$ to an admissible$GP 1$-metric on the triode $T=-1,1\cup0,i$. Another corollary says that atopologically homogeneous $GP 1$-space cannot contain an isometric copy of theinterval $0,1$ and a topological copy of the triode $T$ simultaneously. Alsowe prove that a $GP 1$-metric space $X$ containing an isometric copy of eachcompact $NP 1$-metric space has density not less than continuum.

Author: T. Banakh, D. Repovs, I. Zarichnyi

Source: https://arxiv.org/