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?
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
The equivariant triangulation theorem (Illman 78, Illman 83) says that for $G$ a compact Lie group (for instance a finite group) and $X$ a compact smooth manifold equipped with a smooth $G$-action, there exists a $G$-equivariant triangulation of $X$.
(Illman 72, Thm. 3.1, Illman 83, theorem 7.1, corollary 7.2) Recalled as (ALR 07, theorem 3.2). See also Waner 80, p. 6 who attributes this to Matumoto 71
Moreover, if the manifold does have a boundary, then its G-CW complex may be chosen such that the boundary is a G-subcomplex. (Illman 83, last sentence above theorem 7.1)
These results continue to hold when $G$ is not compact, see Illman00.
Sören Illman, Theorem 3.1 in: Equivariant algebraic topology, Princeton University 1972 (pdf)
Sören Illman, Smooth equivariant triangulations of $G$-manifolds for $G$ a finite group, Math. Ann. 233 (1978) 199-220 [doi:10.1007/BF01405351]
Sören Illman, The Equivariant Triangulation Theorem for Actions of Compact Lie Groups, Mathematische Annalen 262 (1983) 487-502 [dml:163720]
Sören Illman, Existence and uniqueness of equivariant triangulations of smooth proper $G$-manifolds with some applications to equivariant Whitehead torsion, J. Reine Angew. Math. 524 (2000) 129–183. [doi:10.1515/crll.2000.054]
See also
Takao Matumoto, Equivariant K-theory and Fredholm operators, J. Fac. Sci. Tokyo 18 (1971/72), 109-112 (pdf, pdf)
Stefan Waner, Equivariant Homotopy Theory and Milnor’s Theorem, Transactions of the American Mathematical Society Vol. 258, No. 2 (Apr., 1980), pp. 351-368 (JSTOR)
A. Adem, J. Leida and Y. Ruan, Orbifolds and Stringy Topology, Cambridge Tracts in Mathematics 171 (2007) (pdf)
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