Revista de teoría y aplicaciones de la mentira generalizada

On Finite and Infinite Principal Door and Connected-Topologies

Abstract

Azzam AA, Farrag AS and EL-Sanousy E

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.