Continuity in topological groups
For a group \(G\) equipped with a topology \(\tau\), there is much work devoted to finding conditions on the group \(G\) yielding that \(G\) is a topological group. The authors consider groups \(G\) equipped with a Baire metrizable topology \(\tau\) and are interested in finding weaker conditions on the left and right translations that are still sufficient to characterize the group topology. In fact, they obtain the following result as main theorem. Theorem 1. Let \(G\) be a group which is a Baire metrizable space. Suppose that there is a dense subset \(S\) of second category in \(G\) such that the right translations \(\rho_s\) and \(\rho_{s^{-1}}\) are continuous for all \(x\in S\). Suppose further that, for each \(s\in G\), there is a residual subset \(R_s\) of \(G\) such that the left translation \(\lambda_x\) is almost-continuous on \(R_s\), \(\lambda_s(R_x)\) is residual, and \(\lambda_{s^{-1}}\) is almost-continuous on \(\lambda_s(R_x)\). Then \(G\) is a topological group. As an application, they show that if \(G\) is a second countable locally compact right topological group, then its topological center is a topological group.
scientific article; zbMATH DE number 5017445