The object of the paper are the regular topological spaces X in which there exists a metric d related to the topology in the following way: for every nonempty open subset U of X and for every ε>0 there exists a nonempty open subset V of U with d-diameter less than ε. It is shown that such a space X is pseudo-almost Čech complete if, and only if, it contains a dense completely metrizable subset (X is pseudo-almost Čech complete if it is a subset of some pseudocompact space Y and contains as a dense subset some G_δ-subset of Y). A large class of spaces L is described (containing all Borel subsets of compact spaces, all pseudocompact spaces and all p-spaces of Arhangel'skii) such that, if X belongs to L and admits a metric d with the above property, then X is “metrizable up to a first Baire category subset”. I.e. X is the union of two sets X₁ and X₂ where X₁ is of the first Baire category in X and X₂ is metrizable. We provide also different characterizations of the class L.