The Theory Of ω-separation In Lω-spaces

The ω-seperation is one of the most important content in the theory of Lω-spaces. The main content and innovation of this paper are dicussed as follows:1. The concepts of ω-regular and ωT3seperation axioms are introduced. The characterizations of ω-regular and ωT3separation are systematically discussed. Seven equivalent conditions of ω-regular separation are given. That ω-regular and ωT3separations are hereditary, arbitrary multiplicative under condition of full-stratum, topological invariant under (ω1,ω2)-homeomorphic order-homorphism and the good extension in the sense of R. Lowen, are proved.2. The concepts of ω-normal and ωT4seperation axioms are presented, The characterizations of ω-normal and ωT4seperation are systematically studied. That ω-normal and ωT4are hereditary, topological invariant under(ω1,ω2)-homeomorphic order-homorphism and the good extension in the sense of R. Lowen, are obtained. Furthermore, Urysohn lemma in Lω-spaces is proved.3. The concepts of ω-Urysohn seperation axiom, ωθ*-limit point (ωθ*-cluster point) of molecular nets and ideals are defined. The characterizations of ω-Urysohn seperation are systematically discussed. The propeties of ω-Urysohn seperation, such as, the ωθ*-limit point of a molecular net and an ideal is only inω-Urysohn spaces, are obtained. That ω-Urysohn seperation is hereditary, arbitrary multiplicative under condition of full-stratum, topological invariant under(ω1,ω2)-homeomorphic order-homorphism and(ω1,ω2)-homeomorphic order-homorphism, and the good extension in the sense of R. Lowen, are proved.4. The concepts of ω-completely normal and ωT5seperation axioms are established. The characterizations of ω-completely normal and coT5seperation are systematically studied. That ω-completely normal and coT5is hereditary, topological invariant under (ω1,ω2)-homeomorphic order-homorphism and the good extension in the sense of R. Lowen, are given.5. The relations among ωTi(i=-1,0,1,12/1,2,22/1,3,4,5) are presented, which been made to have good implication properties, that is, ωT5(?)ωT4(?)ωT3(?)ωT22/1(?)ωT2(?)ωT12/1(?)ωT1(?)T0(?)ωT-1.