No submaximal topology on a countable set is T-1-complementary

Two T-1-topologies tau and mu given on the same set X, are called transvers al if their union generates the discrete topology on X. The topologies tau and mu are T-1-complementary if they are transversal and their intersection is the cofinite topology on X. We establish that for any connected Tychono ff topology there exists a connected Tychonoff transversal one. Another res ult is that no T-1-complementary topology exists for the maximal topology c onstructed by van Douwen on the rational numbers. This gives a negative ans wer to Problem 162 from Open Problems in Topology (1990).