Cardinal Functions In L-topological Space And A New Separation Axiom

The thesis mainly consist of two parts as following: the first one is thegeneralization - typed studying; the secord one we introduce and study a series ofseparation axioms which is different from those we know before.In the generalization - typed studying, we prove some results still hold on theL - closet topological space which were the important results in paper [3] and [4].We always suppose, throughout the first part, L be a complete lattice. Further-more, we replace the notation molecular by co - prime element which make up of union- genarating sets in complete lattice L. The so - called L - closet topologicalspace, actually, is closed under the operators under arbitrany intersections and finetunions. We also prove those results on Nice - compact in [5] still hold in the case ofhyper - compact and strong - compact topological spaces respectively.In addition, we successfully prove the Hewitt- Marczewski- Pondiczery theoremon L - closet topological space which is an important inequlity in general topology. In the second part, we introduce a series of separation axioms where L - subsetis the object, it is not the same as those separation ones where F - point and closetare the discussing objects. The studying shows, the separation axioms on the onehand can be regarded as a complement to those have existed, on the other hand, itis compatiable in itself, such as T₄(?)T₃(?)T₂, T₁(?)T₀ and so on.