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 ∈ 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 NP n . It is known that a compact metrizable space X has dimension dim ( X ) ⩽ 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 ) → 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 .