Criterion of cyclically compactness in $\sigma$-finite-dimensional Banach modules

Let $X$ be a be a finite-dimensional or a $\sigma$-finite-dimensional Banach $\mathcal{A}$-module, where $\mathcal{A}=C_\infty(Q)$ or $\mathcal{A}=C_\infty(Q)\oplus i\cdot C_\infty(Q)$, $i^2 =-1$, and $C_\infty(Q)$ is an algebra of all continuous functions $f:Q \to [-\infty, +\infty]$ on Stone compact $Q$, taking the values $\pm \infty$ only on nowhere dense sets in $Q$. It is proved that a cyclic set $K$ in $X$ is cyclic compact if and only if $K$ is $C_\infty(Q)$-bounded and $(bo)$-closed.