Noncommutative geometry studies geometric spaces through noncommutative algebras of functions or operators.
For a compact T2-space 
, the commutative C*-algebra 
of complex-valued functions on 
that are continuous determines
up to homeomorphism. Noncommutative geometry reverses
this correspondence and treats a noncommutative C*-algebra
as the algebra of functions on a generalized space that need not have an underlying
set of ordinary points.
Important tools come from operator algebras, K-theory, topology, differential geometry, and measure theory. This point of view extends geometric reasoning to noncommutative algebras.
