  We introduce a new formulation of the so-called topological recursion, that
is defined globally on a compact Riemann surface. We prove that it is
equivalent to the generalized recursion for spectral curves with arbitrary
ramification. Using this global formulation, we also prove that the correlation
functions constructed from the recursion for curves with arbitrary ramification
can be obtained as suitable limits of correlation functions for curves with
only simple ramification. It then follows that they both satisfy the properties
that were originally proved only for curves with simple ramification.
