Consider, however, the space obtained by taking the quotient of two copies A,B of
and glue it and A together along halfspaces so that the singular line of this gluing is transverse in A to the previous singular line.
Call this complicated space K. A branched surface is a space that is locally modeled on K.[1] A branched manifold can have a weight assigned to various of its subspaces; if this is done, the space is often called a weighted branched manifold.
[2] Weights are non-negative real numbers and are assigned to subspaces N that satisfy the following: That is, N is a component of the branched surface minus its branching set.
Weights are assigned so that if a component branches into two other components, then the sum of the weights of the two unidentified halfplanes of that neighborhood is the weight of the identified halfplane.