A subset B of a metric space (X, d) is called a d-bisector set iff there are distinct points x and y in X with B = { z : d ( x , z ) = d ( y , z ) } B = \{ z:d(x,z) = d(y,z)\} . It is shown that if X is a separable metrizable space, then dim ⁡ ( X ) ⩽ n \dim (X) \leqslant n iff X has an admissible metric d for which dim ⁡ ( B ) ⩽ n − 1 \dim (B) \leqslant n - 1 whenever B is a d-bisector set. For separable metrizable spaces, another characterization of n-dimensionality is given as well as a metric dependent characterization of zero dimensionality.