1, The permutation of the finite set is introduced into the denumerable set as well as the definition of the basic rotation, the rotation and the composition of the countability.

2, In this paper, a proof of the theorem on"Any two denumerable Boolean algebras with atoms of the same finite number are isomorphic" is given.

3, In this paper the properties and fixed point theorems of completely continuous maps on denumerable norm spaces are studied.

4, A general model of random walk in time - random environment in any denumerable space is established.

5, A set with cardinality s ＼ s . is said to be denumerable or countably infinite.

6, By using linear operator theory in L2 space, we proved that the operators of this kind has not more than denumerable positive eigenvalues.