Topologies tau(1) and tau(2) on a set X are called T-1-complementary tau(1) boolean AND tau(2) = {X \ F : F subset of X is finite} boolean OR {0} and tau(1) boolean OR tau(2) is a subbase for the discrete topology on X. Topological spaces (X, tau(X)) and (Y, tau(Y)) are called T-1-complementary provided that there exists a bijection f : X --> Y such that tau(X) and {f(-1)(U) : U is an element of tau(Y)} are T-1-complementary topologies on X. We provide an example of a compact Hausdorff space of size 2(c) which is T-1-complementary to itself (c denotes the cardinality of the continuum). We prove that the existence of a compact Hausdorff space of size c that is T-1-complementary to itself is both consistent with and independent of ZFC. On the other hand, we construct in ZFC a countably compact Tikhonov space of size c which is T-1-complementary to itself and a compact Hausdorff space of size c which is T-1-complementary to a countably compact Tikhonov space. The last two examples have the smallest possible size: It is consistent with ZFC that c is the smallest cardinality of an infinite set admitting two Hausdorff T-1-complementary topologies [8]. Our results provide complete solutions to Problems 160 and 161 (both posed by S. Watson [14]) from Open Problems in Topology (North-Holland, 1990).