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?
The continuous group action of a topological group $G$ is called “proper” if it behaves like the action of a compact topological group, without $G$ necessarily being compact.
In particular, the isotropy groups of a proper action are compact.
Consider
a topological group $G$,
a group action of $G$ on $X$ (making $X$ a topological G-space):
The action $\rho \colon (g,x) \mapsto g \cdot x$ is called proper (Palais 61, Def. 1.2.2) if one of the following equivalent conditions hold (their equivalence relies on $X$ being locally compact and Hausdorff, Palais 61, Thm. 1.2.9, Karppinen 16, Rem. 5.2.4):
(Bourbaki properness) The shear map is a proper continuous function:
(Bourbaki 60, Ch. III, Sec. 4.4, review in Lee 00, p. 266)
(Borel properness) For every compact subspace $K \subset X$ the subset
is compact.
(Palais 61, Thm 1.2.9 (5), attributed there to Borel)
(Palais properness) Every point $x \in X$ has a neighbourhood $U_x$ such that every point $y \in U_x$ has a neighbourhood $V_y$ such that
How is the following not a trivial consequence of the fact that compact groups have proper action? Needs clarification.
Let $X$ be a smooth manifold and let $G$ be a compact Lie group. Then every smooth action of $G$ on $X$ is proper.
(e.g. Lee 12, Cor. 7.2)
For more see at equivariant differential topology.
The original definition are due to:
Richard Palais, On the Existence of Slices for Actions of Non-Compact Lie Groups, Annals of Mathematics
Second Series, Vol. 73, No. 2 (Mar., 1961), pp. 295-323 (jstor:1970335, doi:10.2307/1970335, pdf)
(in the context of proving the slice theorem)
Nicolas Bourbaki, Éléments de mathématique, Topologie générale, Chap. 3: Groupes topologiques, 1960
Textbook accounts:
John M. Lee, Section 12.3 of: Introduction to topological manifolds. Graduate Texts in Mathematics 202 (2000), Springer. ISBN: 0-387-98759-2, 0-387-95026-5.
Second edition: Springer, 2011. ISBN: 978-1-4419-7939-1 (doi:10.1007/978-1-4419-7940-7, errata pdf)
John Lee, around p. 147 of: Introduction to Smooth Manifolds, Springer 2012 (doi:10.1007/978-1-4419-9982-5, pdf)
Marja Kankaanrinta, Def. 2.1 in Equivariant collaring, tubular neighbourhood and gluing theorems for proper Lie group actions, Algebr. Geom. Topol. Volume 7, Number 1 (2007), 1-27 (euclid:agt/1513796653)
See also
Further discussion
Sini Karppinen, The existence of slices in $G$-spaces, when $G$ is a Lie group, Helsinki 2016 (hdl:10138/190707)
(in the context of the slice theorem)
Sergey Antonyan, Universal proper G-spaces, Topology and its Applications Volume 117, Issue 1, 15 January 2002, Pages 23-43 (doi:10.1016/S0166-8641(00)00115-2)