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.
 A: It is not really plausible, at least not to me and no more plausible than what it purports to replace namely Caratheodory's "axiom of adiabatic inaccessibility". The latter states that in an arbitrarily small neighborhood of equilibrium there are states that cannot be reached by an adiabatic process.
To your question the answer is in Lieb's paper right after the definition T4 of transversality: 
"To put this in words, the axiom requires that for every adiabat there exists at least one isotherm (i.e., an equivalence class w.r.t. ∼T ), containing points on both sides of the adiabat." 
Recall the standard two-variable (p,V) plots of an ideal gas where you plot the isotherms and the adiabats. They create a net covering the state space and crucially you can define Carnot cycles, for example. You can define them because the isotherms cross the adiabats; when you have more than two variables then you have these adiabatic and isothermic surfaces that fill ("foliate") the state space. 
The entropy function is a strictly monotonic function from one adiabatic surface to another and Caratheodory's axiom is thereby satisfied for a state lying on a lower entropy surface cannot be reached by adiabatic means from a higher entropy surface. The transversality axiom is satisfied by not being able to have a cycle such that one part is isothermic during which energy and entropy are exchanged and while the other is adiabatic with work exchange. That such cycle cannot exist is consistent with Kelvin's axiom of denying the existence of a cycle with a single heat sink.
