Topologically simple group

This article defines a property that can be evaluated for a topological group (usually, a T0 topological group)
View a complete list of such properties
ANALOGY: This is an analogue in topological group of a property encountered in group. Specifically, it is a topological group property analogous to the group property: simple group
View other analogues of simple group | View other analogues in topological groups of group properties (OR, View as a tabulated list)

Definition

Symbol-free definition

A topologically simple group is a topological group satisfying the following equivalent conditions:

1. It no proper nontrivial closed normal subgroup. Note that closedness is purely a property as a subset of the topological space, while normality is a purely group-theoretic property.
2. It has no proper nontrivial quotient group which is a $T_0$-topological group under the quotient topology.
3. Any continuous map from it to a $T_0$-topological group, that is also a group homomorphism, must necessarily be injective.

There may be non-closed normal subgroups, but the corresponding quotient groups will not be $T_0$.

Facts

Closed topological subgroup-defining functions collapse to trivial or improper subgroup

A topological subgroup-defining function is a function that, given a topological group, outputs a unique subgroup of that group. A closed topological subgroup-defining function is a topological subgroup-defining function that always outputs a closed subgroup.

Now we know that the output of a topological subgroup-defining function must be a topologically characteristic subgroup, and hence a normal subgroup. Thus, the output of a closed topological subgroup-defining function must be a closed normal subgroup. In particular, for a topologically simple group, it must be either the whole group, or the trivial subgroup.

Some examples:

• The identity component, viz the connected component containing the identity, must be a closed subgroup. Thus, a topologically simple group is either connected, or totally disconnected (viz, the connected components are one-point subsets).