Given a separable Hilbert space $H$, $U$ is a unitary operator. A cyclic subspace, denoted as $Z(x)$ for some $x\in H$, is defined as the closure of linear span of $U^nx$, where $n\in \Bbb Z$ is any integer number.
Now we have a sequence of cyclic subspaces, namely, $Z(x_1)\subset Z(x_2)\subset \cdots$. Then the closure of its union, $\overline{\bigcup_iZ(x_i)}$, is also a cyclic subspace.