  This work was intended as an attempt to investigate a model of irreversible
process and natural phenomena. For this, we introduce the notion of
irreversible path (that for brevity we write ir-path), ir-homotopy,
ir-contractible space, and Lusternik-Schnirelmann ir-category by equipping the
$I=[0,1]$ with left order topology. We will restrict the irreversibility of
definitions to $T_0$ Spaces, such that for $T_1$ spaces, the ir-paths are
constant. After providing some theorems and properties of these notions,
eventually, we prove that Lusternik-Schnirelmann ir-category is an invariant of
ir-homotopy equivalence.
