On TFU extensions in LCA groups

Let $\mathrm{\ell }$ be the category of all locally compact abelian (LCA) groups. Let $G\in \mathrm{\ell }$ and $H\subseteq G$. The first Ulm subgroup of $G$ is denoted by $G^{(1)}$ and the closure of $H$ by $\bar{H}$. A proper short exact sequence $0\to ABC\to 0$ in $\mathrm{\ell }$ is said to be a $TFU$ extension if $0\to \overline{A^{(1)}}\overline{B^{(1)}}\overline{C^{(1)}}\to 0$ is a proper short exact sequence where $\bar{\phi }=\phi {\mid }_{\overline{A^{(1)}}}$ and $\bar{\psi }=\psi {\mid }_{\overline{B^{(1)}}}$. We introduce some results on $TFU$ extensions. Also, we establish conditions under which the $TFU$ extensions split.