Journal of Generalized Lie Theory and Applications

A space X which carries topology τ is a door space if each subset of X is either open or closed. In this paper a characterization of the principle door and a formula for the number of the door topologies on a set Xn of n points are given. Some properties of the principal connected topologies on non-empty set X are discussed and the minimal τ0 -topologies on X are also characterized. Finally a few results about the number of the chain topologies on Xn are proved.

We use the geometric representation for the Szegö kernel on genus g1+g2 Riemann surface obtained in the frames of the Yamada sewing construction of two Riemann surfaces of genus g1 and genus g2 in order to derive new formulas expressing prime forms. These formulas can be used in vertex operator algebra and conformal field theory computation as well as in algebraic geometry.