Connectedness in some topological vector-lattice groups of sequences - Drewnowski, Lech; Nawrocki, Marek (MATHEMATICA SCANDINAVICA, 2010)
Let $η$ be a strictly positive submeasure on $\mathsf N$. It is shown that the space $ω(η)$ of all real sequences, considered with the topology $τ_{η}$ of convergence in submeasure $η$, is (pathwise) connected iff $η$ is core-nonatomic. Moreover, for an arbitrary submeasure $η$, the connected component of the origin in $ω(η)$ is characterized and shown to be an ideal. Some results of similar nature are also established for general topological vector-lattice groups as well as for the topological vector groups of Banach space valued sequences with the topology $τ_{η}$.