∞-Lie theory (higher geometry)
Background
Smooth structure
Higher groupoids
Lie theory
∞-Lie groupoids
∞-Lie algebroids
Formal Lie groupoids
Cohomology
Homotopy
Related topics
Examples
$\infty$-Lie groupoids
$\infty$-Lie groups
$\infty$-Lie algebroids
$\infty$-Lie algebras
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
Let $G, H \,\in\, Grp(SmthMfd) \xrightarrow{\;Grp(undrl)\;} Grp(TopSp)$ be Lie groups (finite dimensional) with underlying topological groups denoted by the same symbol.
Then every continuous group homomorphism $G \xrightarrow{\;} H$ is smooth.
In other words, the function of hom-sets
is a bijection.
This follows by applying Cartan's closed subgroup theorem to the graph of the homomorphism.
(e.g. Wang 2013, Cor. 1.7, Hall 2015, Cor. 3.50)
Textbook accounts:
Lecture notes:
Last revised on September 3, 2021 at 11:16:31. See the history of this page for a list of all contributions to it.