  For a closed oriented surface $ \Sigma $ we define its degenerations into
singular surfaces that are locally homeomorphic to wedges of disks. Let
$X_{\Sigma,n}$ be the set of isomorphism classes of orientation preserving
$n$-fold branched coverings $ \Sigma\rightarrow S^2 $ of the two-dimensional
sphere. We complete $X_{\Sigma,n}$ with the isomorphism classes of mappings
that cover the sphere by the degenerations of $ \Sigma $. In case $
\Sigma=S^2$, the topology that we define on the obtained completion
$\bar{X}_{\Sigma,n}$ coincides on $X_{S^2,n}$ with the topology induced by the
space of coefficients of rational functions $ P/Q $, where $ P,Q $ are
homogeneous polynomials of degree $ n $ on $ \mathbb{C}\mathrm{P}^1\cong S^2$.
  We prove that $\bar{X}_{\Sigma,n}$ coincides with the
Diaz-Edidin-Natanzon-Turaev compactification of the Hurwitz space
$H(\Sigma,n)\subset X_{\Sigma,n}$ consisting of isomorphism classes of branched
coverings with all critical values being simple.
