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
geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Be?linson-Bernstein localization?
Let $G$ be a compact Lie group and let $V \in RO(G)$ be an orthogonal $G$-linear representation on a real vector space $V$. Then the underlying Euclidean space $\mathbb{R}^V$ inherits the structure of a topological G-space
We may call this the Euclidean G-space associated with the linear representation $V$.
The one-point compactification of a Euclidean $G$-space is the corresponding representation sphere:
Let $V \in RO(G)$ be an orthogonal linear representation of a finite group $G$ on a real vector space $V$.
If $G$ is the point group of a crystallographic group inside the Euclidean group
then the $G$-action on the Euclidean $G$-space $\mathbb{R}^V$ descends to the quotient by the action of the translational normal subgroup lattice $N$ (this Prop.). The resulting $G$-space is an n-torus with $G$-action, which might be called the representation torus of $V$
graphics grabbed from SS 19
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Discussion of equivariant configuration spaces of points in Euclidean $G$-spaces: