There is considered a compactness-continuity result for the solution of a Dirichlet problem as a function of its domain variation. More exactly, the continuity of the following map is studied: \(\Omega \rightarrow u_{\Omega}\) where \(\Omega \subseteq B\) is an open subset of a fixed ball \(B\) in \(\mathbb{R}^{N}\), with \(N \geq 2\), \(f \in H^{-1}(B)\) is a fixed distribution, \(A\) an elliptic operator and \(u_{\Omega}\) is the weak solution of the Dirichlet problem on \(\Omega\) \(-Au_{\Omega}=f,\) \(u_{\Omega |\partial \Omega} =0.\) It is shown that if the variable domain satisfies a uniform Wiener criterion then the map is continuous. An example is given proving that a weak deviation from the uniformity of the Wiener criterion does not preserve the continuity.