The ω-strongly Semi-connected Property And Several Kinds Of Compactness In Lω-Spaces

In2002, many operators are of common property－order-preserving by ProfessorChen Shuili, introduced the concept of L-order-preserving space (L ω-space for short).Hence, some concepts in L-fuzzy topology were extended to L ω-spaces by many demes-tic scholars. Based on these, the author introduces ω-strongly semi-connectedness inLω-spaces, then discusses some properties of the countable ω-compactness inL space, in the last, the countable ω-strongly semi-compactness in L ω-space is studied.The main content of this paper is as follows:1. The concept ofω-strongly semi-open (close) sets is defined and a certainnew connectedness is introduced in Lω-spaces, it is calledω-strongly semi-connecte-dness, and the definition of ω-strongly semi-connected component is given, some oftheir basic properties are studied, at last, it is proved that K Fan’s theorem on strongly semi-connectedness. It is shown thatω-strongly semi-connectedness hasgood properties analogous to the connectedness in general topology.2. The notion of the countable ω-compactness was defined by means of H open covers in L ω-spac, it is proved that countable ω-compactness has many proper-ties. Such as,the sum of finite countable ω-compact sets is a countable ω-compactset, and hereditary with close subset, L-good extension., and it is proved that Tychonoffproduct theorem in L ω-spaces.3. In the last section, the countable ω-strongly semi-compactness in L ω-space isdiscussed. We proved that it is hereditary for ω-strongly semi-closed subsets, and is preserved under the S irresolute mapping, then we give the some characterization ofthe countable ω-strongly semi-compactness.