A locally compact topological group is called almost connected if the underlying topological space of the quotient topological group (of by the connected component of the neutral element, also called the identity component) is compact.
See for instance (Hofmann-Morris, def. 4.24). We remark that since the identity component is closed, the identity in is a closed point. It follows that is and therefore, because it is a uniform space, (a Tychonoff space; see uniform space for details). In particular, is compact Hausdorff.
Also every quotient of an almost connected group is almost connected.
Textbooks with relevant material include
Original articles include