Abstract
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.
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