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
geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
(Arnold-Kuiper-Massey theorem)
The 4-sphere is the quotient space of the complex projective plane by the O(1)-action by complex conjugation (on homogeneous coordinates):
(Arnold 71, Massey 73, Kuiper 74, Arnold 88)
In fact, this is is the beginning of a small pattern indexed by the real normed division algebras:
The 7-sphere is the quotient space of the (right-)quaternionic projective plane by the left multiplication action by U(1) $\subset$ Sp(1):
(Arnold 99, Atiyah-Witten 01, Sec. 5.5)
The 13-sphere is the quotient space of the (right-)octonionic projective plane by the left multiplication action by Sp(1):
The original proof that the 4-sphere is a quotient of the complex projective plane by an action of Z/2:
Vladimir Arnold, On disposition of ovals of real plane algebraic curves, involutions of four-dimensional manifolds and arithmetics of integer quadratic forms, Funct. Anal. and Its Appl., 1971, V. 5, N 3, P. 1-9.
William Massey, The quotient space of the complex projective space under conjugation is a 4-sphere, Geometriae Didactica 1973 (pdf)
Nicolaas Kuiper, The quotient space of $\mathbb{C}P(2)$ by complex conjugation is the 4-sphere, Mathematische Annalen, 1974 (doi:10.1007/BF01432386)
Vladimir Arnold, Ramified covering $\mathbb{C}P^2 \to S^4$, hyperbolicity and projective topology, Siberian Math. Journal 1988, V. 29, N 5, P.36-47
See also
José Seade, Section V.5 in: On the Topology of Isolated Singularities in Analytic Spaces, Progress in Mathematics, Birkhauser 2006 (ISBN:978-3-7643-7395-5)
J. A. Hillman, An explicit formula for a branched covering from $\mathbb{C}P^2$ to $S^4$ (arXiv:1705.05038)
The SO(3)-equivariant enhancement:
The generalization to the 7-sphere being a U(1)-quotient of the quaternionic projective plane is due to
and independently due to
Michael Atiyah, Edward Witten, Section 5.5 of: $M$-Theory dynamics on a manifold of $G_2$-holonomy, Adv. Theor. Math. Phys. 6 (2001) (arXiv:hep-th/0107177, doi:10.4310/ATMP.2002.v6.n1.a1)
(in the context of M-theory on G2-manifolds)
Another proof of these cases and further generalization to the 13-sphere being an Sp(1)-quotient of the octonionic projective plane:
Last revised on September 20, 2021 at 09:58:19. See the history of this page for a list of all contributions to it.