Local connectedness in topological groups
Many results about the conditions for locally compact groups to be locally path-connected were obtained and the interest of the study has turned to more general topological groups, for example, to Polish topological groups. In [Trans. Am. Math. Soc. 212, 393--400 (1975; Zbl 0318.54037)], \textit{G. S. Ungar} showed that every path-connected Polish topological group is locally path-connected. In [Proc. Am. Math. Soc. 130, No. 3, 903--907 (2002; Zbl 0985.54034)], the author showed that every topological group with a second countable, second category path component is locally connected. In this paper, the author gives basic principles underlying these results and, consequently, improves them. For instance, he proves the following results. 1. The following are equivalent for a topological group \(G\): (1) \(G\) is locally connected; (2) \(G\times G\) is locally connected; (3) \(G\) is diagonally connected, where a space \(X\) is \textit{diagonally connected} if for each open set \(V\) in \(X\times X\) containing the diagonal \(D\) of \(X\), there is a connected open set \(W\) such that \(D\subseteq W\subseteq V\); (4) \(G\) contains a subspace \(C\) that is diagonally connected and somewhere dense in \(G\). 2. If a topological group \(G\) is Polish, then the following are equivalent: (1) \(G\) is locally connected; (2) \(G\) is path-connected; (3) \(G\) has a path component that is second category in itself and somewhere dense in \(G\); (4) \(G\) is locally path-connected. Furthermore, if \(G\) is finite dimensional the following statements can be added: (5) \(G\) is locally compact and path-connected; (6) \(G\) is a Lie group. 3. If a topological group \(G\) has a second countable path component that is second category in itself and somewhere dense in \(G\), then \(G\) is almost locally path-connected, and hence locally connected.
scientific article; zbMATH DE number 6383760
scientific article