Module Theory of C*-Algebras

When one first encounters algebraic topology, it is usually in connection with contour integration in complex analysis, or line integrals in multivariable calculus, typically over the plane, or the plane with punctures, or holes. Such integrals are insensitive to small deformations of the curves and suggest studying the curves up to homotopy. Since such regions of the plane are locally simply connected, the local properties of curves from this point of view are unimportant, they are locally homotopic to line segments, but the global properties of the curves can effect values of line integrals, and looking at the ensemble of all curves up to homotopy reveals topological properties of the region.