What is left of CH after you add Cohen reals?

The principle CH∗ concerning elementary submodels is formulated and is shown to be valid in any generic extension obtained by adding any number of Cohen reals to a ground model satisfying CH. s interesting topological consequences, e.g.: 1. ery initially ω1-compact, countably tight T3 space is compact. et X be a countably tight compact T2 space; then 2.1. S is Gδ-dense in X then every point of X is the limit of a converging ω1-sequence from S; Y ⊂ X with s(Y) ⩽ ω1 then h(Y) ⩽ ω1; contains no complete binary tree of closed sets of height ω2. If X is a compact T2 space with small diagonal then X is metrizable.