Branched surface

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]