Noncommutative topology
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Noncommutative topology in mathematics is a term applied to the strictly C*-algebraic part of the noncommutative geometry program. The program has its origins in the Gel'fand duality between the topology of locally compact spaces and the algebraic structure of commutative C*-algebras.
Several topological properties can be formulated as properties for the C*-algebras without making reference to commutativity or the underlying space, and so have an immediate generalization.
Amongst these are compactness (being unital), dimension (real or stable rank), connectedness (projectionless algebra) and K-theory.