Transversal and T1-independent topologies and the Alexandroff duplicate

Two T1-topologies on a given set are called transversal if their union is a subbase for the discrete topology, and T1-independent if their intersection is the cofinite topology. We find new classes of spaces that admit a compact transversal and/or T1-independent topology and present several examples and counterexamples. In Corollary 3.3 we answer a question posed in [I. Juhász, M.G. Tkachenko, V.V. Tkachuk, R.G. Wilson, Self-transversal spaces and their discrete subspaces, Rocky Mountain J. Math. 35 (4) (2005) 1157–1172] in the negative.The Alexandroff duplicate of a topological space plays an important role in our considerations. It proved to be especially useful when constructing topologies which are both transversal to and T1-independent of the topology of a given space.