Complemented topologies on abelian groups

A topology $\tau$ on a group $G$ is complemented if there exists an indiscrete topology $\tau'$ on $G$ such that $U\cap V=\{0\}$ for suitable neighborhoods of zero $U$ and $V$ in the topologies $\tau$ and $\tau'$. The authors give a complementation test for an arbitrary topology. Locally compact groups with complemented topologies have been described. A group all of whose continuous homomorphic images are complete is proved to be compact. A family of $2^\omega$ topologies that are pairwise complementary to one another is defined for an arbitrary group.