CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
A locally compact topological group $G$ is called almost connected if the underlying topological space of the quotient topological group $G/G_0$ (of $G$ 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 $G_0$ is closed, the identity in $G/G_0$ is a closed point. It follows that $G/G_0$ is $T_1$ and therefore, because it is a uniform space, $T_{3 \frac1{2}}$ (a Tychonoff space; see uniform space for details). In particular, $G/G_0$ is compact Hausdorff.
Every compact and every connected topological group is almost connected.
Also every quotient of an almost connected group is almost connected.
Textbooks with relevant material include
M. Stroppel, Locally compact groups, European Math. Soc., (2006)
Karl Hofmann Sidney Morris, The Lie theory of connected pro-Lie groups, Tracts in Mathematics 2, European Mathematical Society, (2000)
Original articles include