Normal decompositions of the lattice of group topologies (Q1079670)

The family L(G) of (not necessarily Hausdorff) topological group topologies on a group G is a complete lattice: For \(A\subseteq L(G)\) the topology \(\vee A\) is the topology on G generated by \(\cup A\), and \(\wedge A=\vee B\) with B the set of topological group topologies \(\tau\) on G such that \(\tau\) \(\subseteq \sigma\) for each \(\sigma\in A.\) For N a normal subgroup of G and \(\tau\in L(N)\), the authors study the semilattice \({\mathcal E}(\tau)\) of L(G) defined by \({\mathcal E}(\tau)=\{\sigma \in L(G):\quad \sigma | N=\tau \}.\) (For suitable G, N and \(\tau\), the relation \({\mathcal E}(\tau)=\emptyset\) is possible.) They show, for example, that if G is abelian with \(G=H_ 1\cdot H_ 2\), and if the topologies \(\tau_ i\) have equal traces on \(H_ 1\cap H_ 2\) \((i=1,2)\), then every \(\sigma\in {\mathcal E}(\tau_ 1)\cap {\mathcal E}(\tau_ 2)\) is connected if both \(\tau_ i\) are connected. Further results describe relations between L(N), L(G/N), and the partition \(\{\) \({\mathcal E}(\tau):\) \(\tau\in L(N)\}\) of L(G).