Let η be a strictly positive submeasure on N. It is shown that the space ω(η) of all real sequences, considered with the topology τη 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 τη.
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