Sequential closure in product spaces

Abstract

Let X denote the product of m-many second countable Hausdorff spaces. Main theorems: (1) If S⊂ X is invariant under compositions, m is weakly accessible (resp., nonmeasurable), and F⊂ S is sequentially closed and a sequential G σ-set which is invariant under projections for finite sets (resp., F⊂ S is sequentially open and sequentially closed), then F is closed. (2) If S⊂ X is invariant under projections and m is nonmeasurable, then every sequentially continuous {0, 1} valued function on S is continuous. (3) A sequentially continuous {0, 1}-valued function on an m-adic space of nonmeasurable weight is continuous. Now let X denote the product of arbitrarily many W-spaces and S⊂ X be invariant under compositions. (4) Then in S, the closure of any Q-open subset coincides with its sequential closure.