Α-fuzzy Compactness in I-topological Spaces

Using a gradation of openness in a (Chang fuzzy) I-topological space, we introduce degrees of compactness that we call α-fuzzy compactness (where α belongs to the unit interval), so extending the concept of compactness due to C. L. Chang. We obtain a Baire category theorem for α-locally compact spaces and construct a one-point α-fuzzy compactification of an I-topological space. 1. Introduction. In 1968, Chang [1] introduced the concept of a fuzzy topol-ogy on a set X. However, some authors criticized that his notion did not really describe fuzziness with respect to the concept of openness of a fuzzy set. In the light of this difficulty, Šostak [9, 10] began his study on fuzzy structures of topological type. Subsequently, by means of some variant of a Šostak fuzzy topology (compare [2]), the authors of [5] developed a theory of α-gradation of open sets (i.e., they introduced the concept of an α-open set where α belongs to the unit interval) for a fuzzy topological space in the sense of Chang. Their theory of gradation of openness makes it possible to introduce degrees of fuzzy topological concepts, which generalize the corresponding ones in general topology on the one hand, and allow one to work with points of X instead of fuzzy points on the other hand. In particular, they proved that the family of all α-neighborhoods (α-nbhds for short) has similar properties as in the classical case; furthermore, they compared their α-T i separation axioms with those discussed in [3]. We would like to draw the attention of the reader to the fact that in the present literature fuzzy topologies in Chang's sense are called I-topologies and gradations of openness are called I-fuzzy topologies (see, e.g., [6, 8]). On the other hand, our study is mainly based on [5] and in the present paper the authors see no need to extend their results to L-(fuzzy) topologies. In our paper, we first introduce a gradation of compactness, namely, α-fuzzy compactness, based both on the aforementioned concept of an α-open set due to [5], as well as, on the notion of compactness due to Chang. Then we investigate the newly defined concepts by establishing analogues of classical topological results related to the concept of compactness. We note that Gantner et al. [4] have introduced a concept of α-compactness based on their notion of an α-shading.