The retraction map $p_{\mathcal{X}}$ satisfies the following properties:

• (i)

  $c_{\mathcal{X}}(x)\in\overline{\{c_{\mathcal{X}}(p_{\mathcal{X}}(x))\}}$ for all $x\in X$.

• (ii)

  $\varphi_{D}\circ p_{\mathcal{X}}=\varphi_{D}$ for all $D\in\Div_{0}(\mathcal{X})_{\mathbf{R}}$.