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
algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
quantum mechanical system, quantum probability
interacting field quantization
(see also Chern-Weil theory, parameterized homotopy theory)
A continous field of $C^\ast$-algebras is a topological space $X$ and a C*-algebra $A_x$ for each point $x \in X$, such that in some sense these algebras vary continuously as $x$ varies in $X$. Hence it is a kind of topological bundle of $C^\ast$-algebras. It is also an example of a Fell Bundle over $X$ when $X$ is thought of as a topological groupoid with only identity morphisms.
In C* algebraic deformation quantization continuous fields of $C^\ast$-algebras over subspaces of the standard interval (tyically $\{1 , \frac{1}{2}, \frac{1}{3}, \cdots, 0\} \hookrightarrow [0,1]$) such that in the limit this becomes a Poisson algebra constitute deformation quantizations of this Poisson algebra.
Last revised on February 25, 2014 at 23:05:44. See the history of this page for a list of all contributions to it.