Noncommutative topology is a recent program having important and deep applications in several branches of mathematics and mathematical physics. Because every commutative -algebra is -isomorphic to where is the space of maximal ideals of (this is the so-called Gelfand theorem) and because an algebraic isomorphism between and induces a homeomorphism between and , -algebraic theory may be regarded as a noncommutative analogue of the algebra of continuous functions vanishing at infinity on a locally compact Hausdorff space. In other words, every property of a locally compact Hausdorff space can be formulated in terms of a "Gelfand dual" property of and then it will probably be true for any noncommutative -algebra.

The following is a list of some such Gelfand dualities (Wegge-Olsen 1993, Moslehian 2002):

topological language	-algebraic language
locally compact Hausdorff space	C-*-algebra
triangulation or the structure of an affine algebraic variety or manifold	system of generators and relations
Stone space	-algebra
sub-Stonean space	-algebra
second countable space	projectionless
Tychonoff product topology	spatial tensor product
proper map	-homomorphism
homeomorphism	automorphism
Radon measure	positive linear functional
compact space	unital
-compact	-unital
compactification	unitization
one-point compactification	minimal unitization
Stone-Cech compactification	maximal unitization (multiplier algebra)
open subset	ideal
closed subset	quotient
clopen	ideal with unit
open dense	essential ideal
complement of singleton	maximal ideal (prime ideal)
discrete	maximal ideals being principal
isolated point	minimal ideal
retract closed subspace	complemented ideal

This entry contributed by Mohammad Sal Moslehian

Moslehian, M. S. "Characterization of Closed Ideals of ." Inter. Math. J. 2, 1055-1059, 2002.

Wegge-Olsen, N. E. K-Theory and C-*-Algebras: A Friendly Approach. Oxford, England: Oxford University Press, 1993.

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Moslehian, Mohammad Sal. "Noncommutative Topology." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/NoncommutativeTopology.html