Orbit sets, transitivity, and sensitivity with upper semicontinuous maps

The paper proves that OF(z) is closed in X^N (Theorem 3.11) and, since X^N is compact, OF(z) is compact; this matches the abstract’s claim. The model gives an alternative route: it first derives that Gr(F) is closed (equivalent to upper semicontinuity for maps into 2^X), then expresses OF(z) as an intersection of preimages of closed sets, and finally proves X^N is compact via a product metric and completeness + total boundedness. These are logically consistent and mutually reinforcing approaches. The paper’s closed-graph equivalence (Proposition 3.3) aligns with the model’s Step 1, and the metric ρ used by the model is the same product metric employed in the paper elsewhere. Thus, both are correct, with different proofs and emphases. See Theorem 3.11 and Proposition 3.3 in the paper, and the abstract’s compactness claim, as well as their use of the metric ρ.