The Theory Of Lω-Hausdorff Expansions

The theory of extensions is one of the most significant research contents of lattice topology.The concepts of LωH-closed space, LωH-set, Lω-minimal Hausdorff space and Lω-Hausdorffexpansion on Lω-space are respectively introduced by using the theory of Moore-SmithLω-convergence and the extensions of Lω-remote neighborhood family, molecular nets, idealsand filter. The theory of extensions of Lω-continuous order homomorphism is used as well. Thebasic theory, equivalent characterizations and applications of these concepts are systemalticallydiscussed. Then the theory of LωH-closed space, Lω-minimal Hausdorff space and the extensiontheory of Lω-Hausdorff space are established. The main research work is as belows:1. The concepts of LωH-set and LωH-closed space are proposed in Lω-space by means of(aω)-remote neighborhood family. The characterizations of LωH-set and LωH-closed space aresystematically discussed. Some important properties of LωH-closed space, such as theLωH-closed space is Lω-regular closed hereditary, arbitrarily multiplicative and preservinginvariance under almost L(ω1, ω2)-continuous order homomorphism are proved.2. The concept of Lω-minimal Hausdorff space is introduced in Lω-space. By using thetheory of Lω-convergence of ideals and molecular nets, the properties of Lω-minimal Hausdorffspace are studied.The condition of a Lω-Hausdorff topology becoming a Lω-minimal Hausdorfftopology is concluded. It is proved that a Lω-Hausdorff space is a Lω-minimal Hausdorff spaceiff every closed ideal having only one Lω-cluster point Lω-converges to it. The topologicalproperties of Lω-minimal Hausdorff space are discussed. Some fundamental properties, such asinverse topological invariant for the Lω-minimal Hausdorff property and so on, are proved. Theconcepts of Lω-semiregular space and Lω-semiregularization are introduced. Finally, therelationships of LωH-closed space and Lω-minimal Hausdorff space are gived.3. The extension theory of Lω-spaces is established. The notions such as Lω-stack, Lω-grill,Lω-trace, Lω-extension and Lω-Hausdorff expansion are gived. Also the order relation andequivalence relation between two Lω-extensions are constructed. Finally, the characterizationsand fundamental properties of Lω-extension and Lω-Hausdorff expansion are discussed.