The Ext group in the categories of topological abelian groups and topological vector spaces

This paper is devoted to the study of the group Ext(G,H) of all extensions of topological abelian groups 0→H→X→G→0 and the group ExtTVS(Z,Y) of all extensions of topological vector spaces 0→Y→X→Z→0. We focus on their behaviour under taking products, countable coproducts, dense subgroups and open subgroups. Finally, we apply the obtained properties to formulate in a more general setting some known results in the category of locally compact abelian groups and to determine conditions in which being a topological vector space is a three space property.