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

This paper is devoted to the study of the group ${\rm Ext} (G,H)$ of all extensions of topological abelian groups $0\to H\to X\to G\to 0$ and the group ${\rm Ext}_{TVS} (Z,Y)$ of all extensions of topological vector spaces $0\to Y\to X\to Z\to 0$. 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.