Boundary tightness and its applications

Given a space X, its  boundary  tightness tb(X) is defined in the same way as tightness if we substitute arbitrary sets by open sets in the definition of tightness. That is, tb(X)= min {kappa: for  any  open  set  U a subset of X we have the equality closure of U= the union of all closures of A where is a subset of U and |A|=<kappa.  
This cardinal  invariant  shows how accessible the boundary of an open set is from a given open set. The  spaces X with tb(X) =<omega will be called spaces with accessible boundaries  of open sets. We will show that by studying the class of spaces with accessible boundaries  of  open sets we can obtain new information about kappa-Frechet--Urysohn spaces, W-spaces in the sense of Gruenhage and even spaces C_p(X).