  In this work, singular surfaces are obtained from smooth orientable closed
surfaces by applying three basic simple loop operations, collapsing operation,
zipping operation and double loop identification, each of which produces
different singular surfaces. A formula that provides the Euler characteristic
of the singularized surface is proved. Also, we introduce a new definition of
genus for singularized surfaces which generalizes the classical definition of
genus in the smooth case. A theorem relating the Euler characteristic to the
genus of the singularized surface is proved.
