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.
Amongst these are compactness (being unital), dimension (real or stable rank), connectedness (projectionless algebra) and K-theory. So we think of a noncommutative C*-algebra as the algebra of functions on a 'noncommutative space' which does not exist classically.
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