Cofinal types and topological groups (Q6611782)

Neighborhoods of the identity of a topological group form a partially ordered set. In the paper, cofinal types of bases of identities of topological groups are studied.\N\NIf \(D\) and \(E\) are two directed sets, then \(D\) is called Tukey reducible to \(E\) if for every unbounded set \(X\subset D\) the image \(f[X]\) is unbounded in \(E\). This is denoted by \(D\leq_T E\). If \(D\leq_T E\) and \(E\leq_T D\), then \(D\) and \(E\) have the same cofinal type, denoted by \(D\equiv_T E\).\N\NIn Section 3, some lemmas about the Tukey order in topological groups are proved:\N\begin{itemize}\N\item[(i)] Let \(G\) be a topological group and \(H\) be a subgroup of \(G\). Then \(H\leq_T G.\)\N\item[(ii)] If \(H\) is an open subgroup of a topological group \(G\). Then \(H\equiv_TG\).\N\item[(iii)] Let \(G\) and \(H\) be topological groups and \(\varphi: G\to H\) be an open and continuous homomorphism. Then \(H\leq_T G.\)\N\item[(iv)] Let \(G\) be a topological group and \(N\triangleleft G\) be a closed normal subgroup of \(G\). Then \(G/N\leq_T G\).\N\end{itemize}\NIn Section 4, the Tukey order for topological groups with some topological restrictions (Fréchet spaces, sequential spaces, tightness) is considered. In Section 6, the equivalence for products of topological groups is studied.