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 Fell Bundle is a family of Banach spaces that varies continuously over the morphism space of a topological groupoid, where nontrivial morphisms in the groupoid induce a bilinear composition across the fibers. They are a generalization of both continuous fields of C*-algebras over topological spaces and of $C^*$-algebraic bundles (fiber bundles of $C^*$-algebras over topological groups).
A Fell Bundle over a groupoid $\mathcal{G}$ is a “continuous” functor $E \colon \mathcal{G} \to\mathfrak{Corr}$, where $\mathfrak{Corr}$ is the $2$-category of $C^*$-algebras and $C^*$- correspondences.
Mulvey considers the special case where the groupoid is a topological space and proves that such Banach space bundles are equivalent to sheaves of Banach spaces, and that both are equivalent to Banach spaces internal to the sheaf topos.
Alex Kumjian, Fell bundles over groupoids, Proceedings of the AMS, volume 126 (1998) (JSTOR)
Alcides Buss, Chenchang Zhu, Ralf Meyer, A higher category approach to twisted actions on $C^*$-algebras, arxiv/0908.0455
There is a variant notion of Fell bundles over inverse semigroups. Those are related to Fell bundles over the corresponding étale groupoids:
Alcides Buss, Ruy Exel, Fell bundles over inverse semigroups and twisted étale groupoids, J. Oper. Theory 67, No. 1, 153-205 (2012) MR2821242 Zbl 1249.46053 arxiv/0903.3388journal; Twisted actions and regular Fell bundles over inverse semigroups, arxiv/1003.0613
Christopher J. Mulvey, Banach sheaves, Journal of Pure and Applied Algebra 17 (1980) 69-83 doi