Two results on spaces with a sharp base

Example – There exists a space X with a sharp base and a perfect mapping onto a space Y which does not have a sharp base.It is 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.Theorem – If X has 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 Fσ set.)A topological 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.