Game approach to universally Kuratowski–Ulam spaces

We consider a version of the open–open game, indicating its connections with universally Kuratowski–Ulam spaces. From [P. Daniels, K. Kunen, H. Zhou, On the open–open game, Fund. Math. 145 (3) (1994) 205–220] and [D. Fremlin, T. Natkaniec, I. Recław, Universally Kuratowski–Ulam spaces, Fund. Math. 165 (3) (2000) 239–247] topological arguments are extracted to show that: Every I-favorable space is universally Kuratowski–Ulam, Theorem 8; If a compact space Y is I-favorable, then the hyperspace exp(Y) with the Vietoris topology is I-favorable, and hence universally Kuratowski–Ulam, Theorems 6 and 9. Notions of uK-U and uK-U∗ spaces are compared.