Connectedness in some topological vector-lattice groups of sequences

Let $\eta$ be a strictly positive submeasure on $\mathsf N$. It is shown that the space $\omega(\eta)$ of all real sequences, considered with the topology $\tau_{\eta}$ of convergence in submeasure $\eta$, is (pathwise) connected iff $\eta$ is core-nonatomic. Moreover, for an arbitrary submeasure $\eta$, the connected component of the origin in $\omega(\eta)$ 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 $\tau_{\eta}$.