The subject matter of topology are discrete invariants of topological spaces (for example smooth manifolds) and maps between them. The simplest such invariant is the winding number of a curve in the plane. One of the principal goals is to study whether a manifold is uniquely determined by its discrete invariants. Of particular interest in the focus subject are stable homotopy theory, K-theory, differential topology, index theory and geometric group theory.
Topology is not an isolated discipline, but has intense connections to other areas of theoretical mathematics, such as algebra, differential geometry, geometric group theory and the theory of operator algebras.