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

A metric space (X,d) has the de Groot property GPn if for any points x0,x1,…,xn+2∈X there are positive indices i,j,k⩽n+2 such that i≠j and d(xi,xj)⩽d(x0,xk). If, in addition, k∈{i,j} then X is said to have the Nagata property NPn. It is known that a compact metrizable space X has dimension dim(X)⩽n iff X has an admissible GPn-metric iff X has an admissible NPn-metric.We prove that an embedding f:(0,1)→X of the interval (0,1)⊂R into a locally connected metric space X with property GP1 (resp. NP1) is open, provided f is an isometric embedding (resp. f has distortion Dist(f)=‖f‖Lip⋅‖f−1‖Lip<2). This implies that the Euclidean metric cannot be extended from the interval [−1,1] to an admissible GP1-metric on the triode T=[−1,1]∪[0,i]. Another corollary says that a topologically homogeneous GP1-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 GP1-metric space X containing an isometric copy of each compact NP1-metric space has density ⩾c.