Sub-separation Axioms In L-topological Spaces And I-fuzzy Topological Spaces

Separation is an essential notion in topology. The topological spaces with many kinds of different separation not only form different spaces classes theoretically, at the same time, it is but also attend by the relevant subjects of topology because of its correlative appliation to other subjects. The aim of this paper is to study the separation in L-topology and I-fuzzy topology, respectively. In L-topology, we introduce the sub-separation axioms including sub-T₁, sub-T₂, sub-T_21/2, sub-T₃ and sub-T₄. In I-fuzzy topology, we establish sub-T₁ and sub-T₂axioms.The paper is organized as follows:The first section is preface. The author introduces background of separation axioms in topology and the development of separation axioms in L-topology and I-fuzzy topology , and gives the problems to be solved.The second section is about sub-separation axioms in L-topological spaces. We introduce the sub-separation axioms including sub-T₁, sub-T₂, sub-T_21/2, sub-T₃ and sub-T₄. And some nice properties of sub-separation axioms are proved. For example, they are hereditary and product invariant, " L-good extension " in Lowen's sense, and the limit of molecular nets is sole under a certain condition for the sub-T₂ space. In addition, the relation between the sub-separation axioms defined in the paper and other separation axioms is also discussed.The third section is about sub-separation axioms in I-fuzzy topological spaces. New sub-T₁ and sub-T₂ axioms are introduced in I-fuzzy topological spaces. Some properties of the new separation axioms are investigated. For ex- ample, they are hereditary and product invariant, and the limit of molecular nets is sole . In addition, we also established the relation between the sub-separation axioms defined in the paper and other separation axioms .