Let \(M\) be a finite-dimensional topological manifold and \(G\) a compact topological group which acts continuously on \(M\). It is well known that \(M\) is metrizable. Let \(\mathcal M\) be the set of all metrics on \(M\) for which the induced topology coincides with the topology of \(M\). Further let \(\mathcal M^G\subseteq\mathcal M\) be the set of \(G\)-invariant metrics and \(\mathcal M^G_0\subseteq\mathcal M\) be the set of \(G\)-invariant complete metrics. The sets \(\mathcal M^G_0\subseteq\mathcal M^G\subseteq\mathcal M\) are endowed with the topology induced by the compact-open topology of \(C(M\times M,\mathbb R)\). In the paper is proved that \(\mathcal M^G\) is a closed subset of \(\mathcal M\) and \(\mathcal M^G_0\) is a dense subset of \(\mathcal M^G\). Moreover, for each compact set \(K\subseteq M\times M\) and \(\delta\in\mathcal M^G\) there is a \(\tilde\delta\in\mathcal M^G_0\) such that \(\tilde\delta|_K=\delta|_K\). Similar results are proved for the sets of \(G\)-invariant Finsler norms and \(G\)-invariant complete Finsler norms on the tangent bundle \(TM\) of \(M\) and the sets of \(G\)-invariant Riemannian metrics and \(G\)-invariant complete Riemannian metrics on \(TM\).