Several Properties Of The Network Order Convergence Topology

In this thesis, we mainly study the properties of order convergence and order topology of net. The full text mainly includes the following three aspects:In the first part, the basic concepts and some results of this thesis are given.In the second part, firstly, we show that the order convergence of net is difference from the convergence order of filter and the order topology of net is difference from the order topology of filter. Sec-ondly, we define the equivalence relation, the quotient sets and the quotient mappings on the complete lattice, and then define the order relation on the quotient sets, which proves that the quotient mapping preserves the upper bound and the lower bound of any subset in the complete lattice. Then we give the base of continuous posets and doubly posets and obtain some equivalent conditions. Finally, we discuss the relation of the order topology of net and interval topolo-gy, Scott topology, Bi-Scott topology and Lawson topology, and we proved that these topologies are equivalent in the linear order sets.In the third part, it is proved that the order convergence of the net on the Cartesian product of the posets and the order convergence on the component posets is equivalent. And we give the defination of continuous mapping with the order topology of net and show a number of equivalent definitions. Finally, we take the whole posets as the object, and the continuous mapping of order preserving with the topology order of net as the morphism form a category, and it is proved that there exist equalizer, finite product and finite coproduct.