# Connected semitopological group

## Definition

A semitopological group is termed **connected** if its underlying topological space is a connected space.

## Facts

- For any semitopological group, there is a closed normal subgroup called the connected component of identity. This is the largest connected subgroup containing the identity element. It is a topological automorphism-invarant subgroup.
- Connected implies no proper open subgroup
- No proper open subgroup not implies connected
- Locally connected and no proper open subgroup implies connected