nLab
closed subgroup

Contents

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Group Theory

Contents

Idea

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.

Definition

Definition

(closed subgroup)

A topological subgroup HGH \subset G of a topological group GG is called a closed subgroup if as a topological subspace it is a closed subspace.

Lemma

(open subgroups of topological groups are closed)

Every open subgroup HGH \subset G of a topological group is closed, hence a closed subgroup.

(e.g Arhangel’skii-Tkachenko 08, theorem 1.3.5)

Proof

The set of HH-cosets is a cover of GG by disjoint open subsets. One of these cosets is HH itself and hence it is the complement of the union of the other cosets, hence the complement of an open subspace, hence closed.

Examples

Stabilizer subgroup

Proposition

(stabilizer subgroups of continuous actions on T1-spaces are closed) Let GG be a topological group and let XGActions(TopologicalSpaces)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 xXx \in X their isotropy groups, hence their stabilizer subgroup

G xStab G(x)G, G_x \;\coloneqq\; Stab_G(x) \;\subset\; G \,,

is a closed subgroup (Def. ).

Proof

We may understand G xG_x as the pre-image

G x(ρ()(x)) 1({x}) G_x \;\simeq\; \big( \rho(-)(x) \big)^{-1} \big( \{x\} \big)

of the singleton subset {x}X\{x\} \subset X under the continuous function which sends xx to its image under the given GG-action:

G ρ()(x) X g gx. \array{ G &\overset{\rho(-)(x)}{\longrightarrow}& X \\ g &\mapsto& g \cdot x \,. }

Since this is a continuous function, and since xXx \in X is a closed point by assumption that XX is T1, hence since {x}X\{x\} \subset X is a closed subset, it follows that G xGG_x \subset G is a closed subset, since continuous preimages of closed subsets are closed.

Localic subgroups

Any localic subgroup of a localic group is closed (see this Theorem).

Properties

For Lie groups

Proposition

(Cartan's closed subgroup theorem)

If HGH \subset G is a closed subgroup of a (finite dimensional) Lie group, then HH 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.

Local sections over coset space

Proposition

(coset space coprojections with local sections)
Let GG be a topological group and HGH \subset G a subgroup.

Sufficient conditions for the coset space coprojection GqG/HG \overset{q}{\to} G/H to admit local sections, in that there is an open cover iIU iG/H\underset{i \in I}{\sqcup}U_i \to G/H and a continuous section σ 𝒰\sigma_{\mathcal{U}} of the pullback of qq to the cover,

G |𝒰 G σ (pb) iIU i = iIU i G/H, \array{ && G_{\vert \mathcal{U}} &\longrightarrow& G \\ & {}^{\mathllap{ \exists \sigma }} \nearrow & \big\downarrow &{}^{{}_{(pb)}}& \big\downarrow \\ \mathllap{ \exists \; } \underset{i \in I}{\sqcup} U_i &=& \underset{i \in I}{\sqcup} U_i &\longrightarrow& G/H \mathrlap{\,,} }

include the following:

or:

References

Last revised on April 12, 2021 at 08:12:43. See the history of this page for a list of all contributions to it.