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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, lowerseparation axioms are interpreted into the condition for foliations and severalrelations of them are described. Moreover, we introduce some notions fortopologies from dynamical systems and foliation theory.

Preorder characterizations of lower separation axioms and their applications to foliations and flows