A surface is a space that locally looks like ${\displaystyle \mathbb {R} ^{2}}$  (a Euclidean space, up to homeomorphism).

Consider, however, the space obtained by taking the quotient of two copies A,B of ${\displaystyle \mathbb {R} ^{2}}$  under the identification of a closed half-space of each with a closed half-space of the other. This will be a surface except along a single line. Now, pick another copy C of ${\displaystyle \mathbb {R} }$  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:

• N is open.
• N does not include any points whose only neighborhoods are the quotient space described above.
• N is maximal with respect to the above two conditions.

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.

• Branched covering
• Branched manifold