On metric spaces with the properties of de Groot and Nagata in dimension one (original) (raw)

2010, Topology and its Applications

A metric space (X, d) has the de Groot property GPn if for any points x 0 , x 1 ,. .. , x n+2 ∈ X there are positive indices i, j, k ≤ n + 2 such that i = j and d(x i , x j) ≤ d(x 0 , x k). 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 GP 1 (resp. NP 1) 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 GP 1-metric on the triode T = [−1, 1]∪ [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 ≥ c.