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