(geometry Isbell duality algebra)
topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
higher geometry / derived geometry
Ingredients
Concepts
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
Constructions
Examples
derived smooth geometry
Theorems
symmetric monoidal (∞,1)-category of spectra
noncommutative topology, noncommutative geometry
noncommutative stable homotopy theory
genus, orientation in generalized cohomology
By Gelfand duality, suitable topological spaces are contravariantly equivalent to commutative C*-algebras. Therefore conversely, non-commutative -algebras may be thought as the formal duals of generalized topological spaces, “noncommutative topological spaces”. Therefore the study of operator algebra and C-star-algebra theory is sometimes called noncommutative topology. This is a special case of the general idea of noncommutative geometry.
(Not to be confused with algebraic topology, which is instead the study of ordinary topology and of its homotopy theory by algebraic tools.)
duality between algebra and geometry
in physics:
geometric context | universal additive bivariant (preserves split exact sequences) | universal localizing bivariant (preserves all exact sequences in the middle) | universal additive invariant | universal localizing invariant |
---|---|---|---|---|
noncommutative algebraic geometry | noncommutative motives | noncommutative motives | algebraic K-theory | non-connective algebraic K-theory |
noncommutative topology | KK-theory | E-theory | operator K-theory | … |
Last revised on November 15, 2021 at 15:40:15. See the history of this page for a list of all contributions to it.