Topologically simple group

Symbol-free definition
A topologically simple group is a defining ingredient::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 $$T_0$$.

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).