In this paper we define and examine structures $(V,B,\in,d)$, where $(V,B,\in)$ is a $t$-design or $[n,n+m]$-net, $d$ is a mapping from $V^t$ to $R\backslash R^-$ which satisfies certain axioms analogous to the axioms of usual metrics. The mapping $d$ is a generalization of the usual metrics so we call it $n$-metric function and we call the structure $(V,B,\in,d)$ $n$-metric space. Then we define a $t$-induced topology and examine it