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
A real Lie group is compact if its underlying topological group is a compact topological group.
Compact Lie groups have a very well understood structure theory.
All maximal tori of a compact Lie group are conjugate by inner automorphisms. The dimension of a maximal torus of a compact Lie group is called the rank of . The normalizer of a maximal torus determines . The Weyl group of with respect to a choice of a maximal torus is the group of automorphisms of which are restrictions of inner automorphisms of . The maximal torus is of finite index in its normalizer; the quotient is isomorphic to . The cardinality of for a compact connected , equals the Euler characteristic of the homogeneous space (“flag variety”).
See also at relation between compact Lie groups and reductive algebraic groups
Last revised on August 26, 2019 at 07:48:31. See the history of this page for a list of all contributions to it.