  In this paper, we characterize several lower separation axioms $C_0, C_D$,
$C_R$, $C_N$, $\lambda$-space, nested, $S_{YS}$, $S_{YY}$, $S_{YS}$, and
$S_{\delta}$ using pre-order. To analyze topological properties of (resp.
dynamical systems) foliations, we introduce notions of topology (resp.
dynamical systems) for foliations. Then proper (resp. compact, minimal,
recurrent) foliations are characterized by separation axioms. Conversely, lower
separation axioms are interpreted into the condition for foliations and several
relations of them are described. Moreover, we introduce some notions for
topologies from dynamical systems and foliation theory.
