Two results on spaces with a sharp base

Example exists a space X with a sharp base and a perfect mapping f : X → Y onto a space Y which does not have a sharp base. known that a spaces with a sharp base have a point-countable sharp base. This can be sharpened to “point-finite” on the set of isolated points. m as a sharp base then X has a point-countable sharp base which is point-finite on the set Z of isolated points. (Hence Z is an F σ set.) logical proof of the previous theorem is given but the theorem follows from a more general combinatorial statement about certain subsets of κ × κ . This last statement is proved using a set-theoretic argument.