Regular G-diagonals and hyperspaces

In this paper, we mainly prove that open k-to-one maps need not inversely preserve regular Gδ-diagonals (or zero-set diagonals), and open k-to-one maps preserve zero-set diagonals. If X is continuously semi-metrizable, then the Vietoris hyperspace F(X) is also continuously semi-metrizable. For a Hausdorff space X, the Pixley-Roy hyperspace F2[X] has a regular Gδ-diagonal if and only if each point of X is a Gδ-set.