Characterizations of open and semi-open maps of compact Hausdorff spaces by induced maps

Let f X→ Y be a continuous surjection of compact Hausdorff spaces. By f_*(X)(Y),\ μ↦ μ∘ f⁻¹ and 2ᶠ2X→2Y,\ A↦ f[A] we denote the induced continuous surjections on the probability measure spaces and hyperspaces, respectively. In this paper we mainly show the following facts: (1) If f_* is semi-open, then f is semi-open. (2) If f is semi-open densely open, then f_* is semi-open densely open. (3) f is open iff 2ᶠ is open. (4) f is semi-open iff 2ᶠ is semi-open. (5) f is irreducible iff 2ᶠ is irreducible.