Branched surface

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In mathematics, a branched surface is type of topological space. A small piece of an surface looks topologically (i.e., up to homeomorphism) like ℝ². A small piece of a branched surface, on the other hand, might look like either of the following:

• ℝ²;
• the quotient space of two copies of ℝ² modulo the identification of a closed half-space of each with a closed half-space of the other. Needs work.

A neighborhood in a branched surface which cannot appear in a surface

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. 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 the space from one branching to the next. Weights are assigned so that any if a neighborhood of a point is the quotient space described above, then the sum of the weights of the two unidentified hyperplanes of that neighborhood is the weight of the identified hyperplane space.