Removable sets for the Newtonian spaces $N^{1,p}$

The pattern developed by Vodop'yanov and Gol'dstein is used to introduce suitable $NC_p$-sets in a domain $Q$ of a metric measure space $X$ with $p$-locally bounded geometry. Criteria of identity are found for the Newtonian spaces $N^{1,p}(Q\setminus E)$ and $N^{1,p}(Q)$ (for the Dirichlet spaces $D^p(Q\setminus E)$ and $D^p(Q)$, respectively) in terms of $E$ as an $NC_p$-set in $Q$, $1<p<\infty$. From this it is deduced that domains $Q$ and $Q_1$, $Q_1 \subset Q$, are $(1,p)$-equivalent if and only if $Q\setminus Q_1$ is an $NC_p$-set in $Q$. Moreover, the completeness of $D^p(Q)$ is proved and, for a quasisymmetric map $f:X\to Y$ of two locally $\mathbf Q$-regular metric spaces $X$ and $Y$ with $\mathbf Q$-locally bounded geometry, it is shown that $f(E)$ is an $NC_{\mathbf Q}$-set in $f(Q)$ if and only if $E$ is an $NC_{\mathbf Q}$-set in $Q\subset X$.