Parametric topological entropy of possibly discontinuous maps in compact Hausdorff spaces and hyperspaces

The paper’s Theorem 5.2 proves exactly the chain hspan_ρ ≤ hsep_ρ ≤ hH ≤ h(ϕ∗) for u.s.c. multivalued dynamics on compact uniform spaces, by combining Lemma 3.19 (maximal separated implies spanning), the definition/equality framework for the H-entropy, and Proposition 5.1 (embedding X as singletons in K(X)) to obtain hH ≤ h(ϕ∗). The candidate solution reproduces these same ingredients with essentially the same logical steps: (1) maximal separated ⇒ spanning (ρ), (2) pH ≥ pρ ⇒ hsep_ρ ≤ hH, and (3) singleton embedding iota plus (ϕ∗)[k]({x}) = ϕ[k](x) ⇒ hH ≤ h(ϕ∗). The small differences are stylistic (e.g., an explicit appeal to pH ≥ pρ and Zorn’s lemma for maximal sets), not substantive. Thus both are correct and follow substantially the same proof structure.