  A sequence $\{a_n\}$ in a group $G$ is a {\em $T$-sequence} if there is a
Hausdorff group topology $\tau$ on $G$ such that
$a_n\stackrel\tau\longrightarrow 0$. In this paper, we provide several
sufficient conditions for a sequence in an abelian group to be a $T$-sequence,
and investigate special sequences in the Pr\"ufer groups
$\mathbb{Z}(p^\infty)$. We show that for $p\neq 2$, there is a Hausdorff group
topology $\tau$ on $\mathbb{Z}(p^\infty)$ that is determined by a $T$-sequence,
which is close to being maximally almost-periodic--in other words, the von
Neumann radical $\mathbf{n}(\mathbb{Z}(p^\infty),\tau)$ is a non-trivial finite
subgroup. In particular, $\mathbf{n}(\mathbf{n}(\mathbb{Z}(p^\infty),\tau))
\subsetneq \mathbf{n}(\mathbb{Z}(p^\infty),\tau)$. We also prove that the
direct sum of any infinite family of finite abelian groups admits a group
topology determined by a $T$-sequence with non-trivial finite von Neumann
radical.
