  We demonstrate a striking effect of non-commutative (NC) geometry on
topological properties of gauge theory by Monte Carlo simulations. We study 2d
U(1) NC gauge theory for various boundary conditions using a new finite-matrix
formulation proposed recently. We find that a single topological sector
dictated by the boundary condition dominates in the continuum limit. This is in
sharp contrast to the results in commutative space-time based on lattice gauge
theory, where all topological sectors appear with certain weights in the
continuum limit. We discuss possible implications of this effect in the context
of string theory compactifications and in field theory contexts.
