Abstract
Let f:X→Y be a continuous surjection of compact Hausdorff spaces. Byf⁎:M(X)→M(Y),μ↦μ∘f−1 and 2f:2X→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.
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