Abstract
We present a survey on transversal, T1-independent, and T1-complementary topologies on sets and groups. First we consider the complementarity problem in the respective lattices L(X) and L1(X) of all topologies and T1-topologies on an infinite set X. The second part of the article deals with the complementarity problem in the family G2(G) of Hausdorff topological group topologies on a given infinite group G. It turns out that NO element of G2(G) has a complement in G2(G), so the main attention is paid to the study of transversality and T1-independence in G2(G).The article contains well-known old results, recent results dispersed in many sources, and several new facts obtained by the authors of this contribution. Selected open problems are included here to indicate that this area of research is far from complete.
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