The Research Of Countability Axioms In I-fuzzy Topological Spaces

Countability is a fundamental notion in topology. The topological spaces with many kinds of different countabilities not only theoretically form different space classes, at the same time, it is but also attened by the relevant subjects of topology because of its correlative appliation to other subjects. The aim of this paper is to study the countability on I-fuzzy topological spaces.The first section is preface. The author introduces background about generation of countability in topology and status about the research and the development of count-ability on lattice-valued topological spaces, and gives the main problems to be soled.The second section is the preparation. The author introduces some notations and simply reviews some fundamental knowledge about I-fuzzy topological spaces.The third section is the research about I-fuzzy first countability and I-fuzzy second countability in I-fuzzy topological spaces. The author uses I-fuzzy quasi-coincident neighborhood system to introduce the concepts of I-fuzzy first countability and I-fuzzy second countability, and proves the degree to second countability in each of I-fuzzy topological space must be less than that to first countability, and gives the equivalent characterations about the continuity of function by convergence of sequences in first countable I-fuzzy topological spaces.The fourth section is the reseach about I-fuzzy separability and I-fuzzy Lindelof property. The author introduces the concepts of I-fuzzy separability and I-fuzzy Lindelof property, and proves that the degree to I-fuzzy second countability in each of I-fuzzy topological spaces must be less than that to I-fuzzy separability and I-fuzzy Lindelof property.The fifth section is the reseach about countable productive property of I-fuzzy first countability in I-fuzzy topological space. The author proves the infimum of the degree to I-fuzzy first countability in each of I-fuzzy topilogical spaces is less than that in their productive space.The sixth section is remark. The author proposes some problems for further study.