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
vector bundle, 2-vector bundle, (∞,1)-vector bundle
real, complex/holomorphic, quaternionic
A continous field of $C^\ast$-algebras is a kind of topological bundle of $C^\ast$-algebras: A topological space $X$ and a C*-algebra $A_x$ for each point $x \in X$, such that these algebras vary continuously, in some sense, as $x$ varies in $X$. When $X$ is thought of as a topological groupoid with only identity morphisms, this may be understaood as a special case of Fell bundles over $X$.
In C* algebraic deformation quantization continuous fields of $C^\ast$-algebras over subspaces of the standard interval (tyically $\{1 , \frac{1}{2}, \frac{1}{3}, \cdots, 0\} \hookrightarrow [0,1]$) which at $\hbar = 0$ are Poisson algebras constitute non-perturbative deformation quantizations of this Poisson algebra (hence of the phase space of some physical system that it represents).
The notion originates with:
An efficient reformulation is due to:
See also:
Last revised on January 15, 2024 at 03:53:29. See the history of this page for a list of all contributions to it.