Let $\succsim$ a complete, reflexive and transitive binary relation defined on $X$, a non-degenerated (i.e not identical to a singleton) convex compact subset of $\mathbb{R}^n_{++}$ (the set of n-dimensional vectors with positive components). Suppose $\succsim$ is continuous for the order topology and strictly monotonic (i.e, it preserves the usual partial order of $\mathbb{R}^n_{++}$).
Are the associated indifference sets connected ?
For a preference relation defined on the whole convex cone $\mathbb{R}^n_{++}$ the properties I introduced are indeed sufficient, as shown in this post Are Indifference Curve graphs continuous given the preferential definition of continuity?
Can you think of a counter-example or, on the contrary, provide the proof of the sufficiency of these conditions ?
Thank you very much !