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
symmetric monoidal (∞,1)-category of spectra
(geometry $\leftarrow$ Isbell duality $\to$ algebra)
A dense subalgebra of a topological algebra is a subalgebra? which is also a dense subspace.
For instance, the algebra of smooth functions on a given smooth manifold $M$ is a dense subalgebra of the C-star-algebra of continuous functions on $X$. Conversely, if we start with a commutative C-star-algebra $A$ and view it as the algebra of continuous maps on its spectrum $S$, then picking a dense subalgebra of $A$ may specify a smooth structure on $S$ (which may be any compact Hausdorff space). For noncommutative geometry, we can let $A$ be an arbitrary $C^*$-algebra. One also speaks of smooth C-star algebras if a $C^\ast$-algebra is equipped with an inverse system of inclusions of dense subalgebra (a noncommutative version of Frechet spaces).
Hence equipping a C-star-algebra with a choice of dense subalgebra serves may be thought of as refining from noncommutative topology to noncommutative geometry. This is for instance what happens in the definition of spectral triples and in smooth refinements of KK-theory.
E. Kissin, V. S. Shulman, Differential properties of some dense subalgebras of $C^\ast$-algebras, Proceedings of The Edinburgh Mathematical Society, vol. 37, no. 03 (1994)
Larry Schweitzer, Dense nuclear Fréchet ideals in $C^\ast$-algebras (arXiv:1205.0089)
Last revised on May 31, 2022 at 16:00:55. See the history of this page for a list of all contributions to it.