simply connected region

For the purpose of the theorem every "country" has to be a simply connected region, or contiguous.

Equivalently, we can work in two simply connected regions with cuts that pass from the tube towards or away from the detection screen.

In any simply connected region outside of the tube we can find a Gauge theory (acting on wave functions and connections) that gauges away the vector potential.

The entanglement entropy of a simply connected region of boundary length L, within an infinite two-dimensional topologically ordered state, has the following form for large L:

Then there is a simply connected region G contained with its closure in the unit disc, such that f maps G onto one of the regions D homeomorphically.

A similar result holds for a harmonic function f on a simply connected region with smooth boundary provided the L norms are taken over the level curves in a tubular neighbourhood of the boundary.