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
Classical groups
Finite groups
Group schemes
Topological groups
Lie groups
Super-Lie groups
Higher groups
Cohomology and Extensions
Related concepts
A common condition on subgroups of a topological group is that they be closed.
This is typically required in equivariant differential topology/equivariant homotopy theory, for instance in the definition of the orbit category.
(closed subgroup)
A topological subgroup $H \subset G$ of a topological group $G$ is called a closed subgroup if as a topological subspace it is a closed subspace.
(open subgroups of topological groups are closed)
Every open subgroup $H \subset G$ of a topological group is closed, hence a closed subgroup.
(e.g Arhangel’skii-Tkachenko 08, theorem 1.3.5)
The set of $H$-cosets is a cover of $G$ by disjoint open subsets. One of these cosets is $H$ itself and hence it is the complement of the union of the other cosets, hence the complement of an open subspace, hence closed.
(stabilizer subgroups of continuous actions on T1-spaces are closed) Let $G$ be a topological group and let $X \in G Actions(TopologicalSpaces)$ a topological G-space whose underlying topological space is a T1-space, e.g. a Hausdorff space.
Then for all points $x \in X$ their isotropy groups, hence their stabilizer subgroup
We may understand $G_x$ as the pre-image
of the singleton subset $\{x\} \subset X$ under the continuous function which sends $x$ to its image under the given $G$-action:
Since this is a continuous function, and since $x \in X$ is a closed point by assumption that $X$ is T1, hence since $\{x\} \subset X$ is a closed subset, it follows that $G_x \subset G$ is a closed subset, since continuous preimages of closed subsets are closed.
Any localic subgroup of a localic group is closed (see this Theorem).
(Cartan's closed subgroup theorem)
If $H \subset G$ is a closed subgroup of a (finite dimensional) Lie group, then $H$ is a sub-Lie group, hence a smooth submanifold such that its group operations are smooth functions with respect to the the submanifold smooth structure.
(coset space coprojections with local sections)
Let $G$ be a topological group and $H \subset G$ a subgroup.
Sufficient conditions for the coset space coprojection $G \overset{q}{\to} G/H$ to admit local sections, in that there is an open cover $\underset{i \in I}{\sqcup}U_i \to G/H$ and a continuous section $\sigma_{\mathcal{U}}$ of the pullback of $q$ to the cover,
include the following:
$G$ is any topological group
and $H$ is a compact Lie group
(in particular for $H$ a closed subgroup if $G$ itself is a compact Lie group, since closed subspaces of compact Hausdorff spaces are equivalently compact subspaces).
or:
The underlying topological space of $G$ is
(e.g. if $G$ is a Lie group)
and $H \subset G$ is a closed subgroup.
Andrew Gleason, Spaces With a Compact Lie Group of Transformations, Proceedings of the American Mathematical Society Vol. 1, No. 1 (Feb., 1950), pp. 35-43 (jstor:2032430, doi:10.2307/2032430)
Paul Mostert, Local Cross Sections in Locally Compact Groups, Proceedings of the American Mathematical Society, Vol. 4, No. 4 (Aug., 1953), pp.645-649 (jstor:2032540, doi:10.2307/2032540)
Alexander Arhangel’skii, Mikhail Tkachenko, Topological Groups and Related Structures, Atlantis Press 2008 (doi:10.2991/978-94-91216-35-0)
Last revised on September 4, 2021 at 18:16:22. See the history of this page for a list of all contributions to it.