We prove that every connected locally compact Abelian topological group is sequentially connected, i.e., it cannot be the union of two proper disjoint sequentially closed subsets. This fact is then applied to the study of extensions of topological groups. We show, in particular, that if $H$ is a connected locally compact Abelian subgroup of a Hausdorff topological group $G$ and the quotient space $G/H$ is sequentially connected, then so is $G$.

Various techniques are presented for constructing $\Lambda$ (p) sets which are not $\Lambda (p+ϵ)$ for all $ϵ\>0$. The main result is that there is a $\Lambda$ (4) set in the dual of any compact abelian group which is not $\Lambda (4+ϵ)$ for all $ϵ\>0$. Along the way to proving this, new constructions are given in dual groups in which constructions were already known of $\Lambda$ (p) not $\Lambda (p+ϵ)$ sets, for certain values of $p$. The main new constructions in specific dual groups are:– there is a $\Lambda$ (2k) set which is not $\Lambda (2k+ϵ)$ in $\mathbf{Z}\left(2\right)\oplus \mathbf{Z}\left(2\right)\oplus \cdots$ for all $2\le k$, $k\in \mathbf{N}$ and $ϵ\>0$, and in $\mathbf{Z}\left(p\right)\oplus \mathbf{Z}\left(p\right)\oplus \cdots$ ($p$ a prime,...