The Strongω_α Continuity, ω_α-Separations And O_r-Countability On L_ω-Spaces

By the basis of ω_α-stratified order-preserving operator, ω_α-closed andω_α-continuity are introduced in paper [2].And their basic properties are discussed.Based on the notions, the strong ω α continuity, ω_α separations andO_r-countabilityare given in L_ω spaces. The main research contents and results are as follows:1. In L ω spaces, the concept of O_r-open set is given based on ω_α-closed. Thenthe strong ω_α-continuity and ω_α-order homomorphism are defined based onω_α-closed set. Their properties are discussed and the relationships between strongω_α-continuity and ω_α-continuity are given.2. The concepts ofω_α-T0,ω_α-T-1,ω_α-T1,ω_α-T2separations are given firstly inL_ω spaces. Then the hereditary property, countable multiplicative property and therelationships of them are discussed.3. In L_ω spaces, the O_r-base, O_r-subbase and O_r-remote neighborhood baseare introduced. The concepts of first and second O_r-countably spaces are defined withthe help of O_r-base and O_r-remote neighborhood base.The relationships between thetwo countability properties are discussed. Both of the first and second O_r-countabilityare hereditary property, countable multiplicative property and invariable property underO_r-order homomorphism of homomorphism is proved; the concept of the quasiω..α-Lindelofproperty is defined based on ω_α-closed set. Therefore, the main resultsof the quasi ω_α-Lindelo..fproperty in L ω order-preserving operator spaces arepreserved in ω_α-order-preserving operator spaces.