AQFT and operator algebra
homotopical algebraic quantum field theory
topology (point-set topology, point-free topology)
see also 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
The Gelfand spectrum (originally Гельфанд) of a commutative C*-algebra $A$ is a topological space $X$ such that $A$ is the algebra of complex-valued continuous functions on $X$. (This “Gelfand duality” is a special case of the general duality between spaces and their algebras of functions.)
Given a unital, not necessarily commutative, complex C*-algebra $A$, the set of its characters, that is: continuous nonzero linear homomorphisms into the field of complex numbers, is canonically equipped with what is called the spectral topology which is compact Hausdorff. (If applied to a nonunital $C^*$-algebra, then it is only locally compact.) This correspondence extends to a functor, called the Gel’fand spectrum from the category C*Alg of unital $C^*$-algebras to the category of Hausdorff topological spaces.
A character on a unital Banach algebra is automatically a continuous function (with Lipschitz constant 1).
The Gelfand spectrum functor is a full and faithful functor when restricted to the subcategory of commutative unital $C^*$-algebras.
The kernel of a character is clearly a codimension-$1$ closed subspace, and in particular a closed maximal ideal in $A$; therefore the Gel’fand spectrum is a topologised analogue of the maximal spectrum of a discrete algebra.
For noncommutative $C^*$-algebras the spaces of equivalence classes of irreducible representations (i.e., the spectrum) and their kernels (i.e., the primitive ideal space) are more important than the character space.
The Gelfand spectrum is also useful in the context of more general commutative Banach algebras.