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

TL;DR

This paper characterizes various lower separation axioms using pre-order theory and applies these characterizations to analyze properties of foliations and dynamical systems, introducing new notions for topologies in these contexts.

Contribution

It provides novel pre-order characterizations of lower separation axioms and applies them to the study of foliations and dynamical systems, establishing new links between topology and dynamics.

Findings

Characterization of separation axioms via pre-order.

Application to properties of foliations and dynamical systems.

Introduction of new topological notions for foliations and dynamics.

Abstract

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.