Weakly locally compact topological Abelian groups and their basic properties

The notion of a weakly locally compact topological abelian group introduced in this paper generalizes the notion of a fibrous topological abelian group studied by N. Ya. Vilenkin.
Since in the class of locally compact topological abelian groups we distinguish classes of compact and discrete topological groups, in the class of weakly locally compact topological abelian groups we distinguish classes of quasicompact and quasidiscrete groups which are dual in the sense of Pontryagin's theory of characters. We prove here that the group of characters of a weakly locally compact topological abelian group is weakly locally compact and construct universal groups for weakly locally compact groups.