On the realization of the metric-dependent dimension function $d_{2}$

Abstract:

Let $(X,\rho )$ be a metric space with ${d_2}(X,\rho ) < \dim X$ where ${d_2}$ denotes the metric-dependent dimension function introduced by K. Nagami and J. H. Roberts [2]. Then it will be shown that for any integer $k$ with ${d_2}(X,\rho ) \leq k \leq \dim X$ there exists a topologically equivalent metric ${\rho _k}$ with ${d_2}(X,{\rho _k}) = k$. This extends a result of J. C. Nichols [3] and answers the problem raised by K. Nagami and J. H. Roberts [2] in the affirmative.