# The transversality axiom in Lieb's Thermodynamics paper

I'm reading The Physics and Mathematics of the Second Law of Thermodynamics and have a question about the T4 transversality axiom which is writtern on page 54.

T4) Transversality. if $$\Gamma$$ is the state space of a simple system and if $$X \in \Gamma$$, then there exist states $$X_0 \overset{T}{\sim}X_1$$ with $$X_0 \prec\prec X \prec\prec X_1.$$

The relation $$\prec$$ is the adiabatic accessibility. $$X\prec Y$$ means that there is an adiabatic transition from $$X$$ to $$Y$$. $$X \prec \prec Y$$ means $$X\prec Y$$ and $$Y \not\prec X$$.

The relation $$\overset{T}{\sim}$$ is the thermal equilibrium.

Why is this axiom plausible?

I don't know orthodox thermodynamics and started studying from this paper.