Sets In Fuzzy Topological Spaces Research Articles

On nearly compact and θ-rigid fuzzy sets in fuzzy topological spaces

Abstract This paper intends to initiate the study of nearly compact and θ-rigid fuzzy sets in a fuzzy topological space. These two classes of fuzzy sets are seen to be properly contained in the class of fuzzy compact sets and properly contain the class of fuzzy almost compact sets. Several characterizations and certain other properties of these two types of fuzzy sets are established by using the concepts of fuzzy δ-closures and θ-closures of fuzzy sets and fuzzy filterbases.

Almost compact fuzzy sets in fuzzy topological spaces

The study of almost compact fuzzy topological spaces was initiated by Concilio and Gerla [4], which was followed by further investigations of the same in [11]. The present paper aspires for an extension of the same concept to fuzzy sets in a fuzzy topological space. We characterize almost compact fuzzy sets in terms of the concept of fuzzy filterbase. Some properties of such sets are established using the concepts of fuzzy ds-closure and fuzzy ds-cluster points of fuzzy filterbase.

Separations in lattice-valued induced spaces

For the value set (i.e. range) I, the induced space is normal if and only if its underlying space is normal. But for a general value set, the situation is quite different. In this paper, we construct a skillful example to show that the corresponding relation about the normalities of these two spaces does not hold for a certain fuzzy lattice. Meanwhile, for an important type of fuzzy lattices, using the new method, we successfully establish a positive result about this normality relation. Furthermore, we shall describe an α-level set of the closure (or interior) of a fuzzy set A in terms of the topology of the underlying space and some level sets of A. This construction problem of closed sets and open sets in fuzzy topological spaces possesses independent interest. Other separation axioms, such as Hausdorff separation and regularity, are also discussed in this paper.

A note on induced and coinduced fuzzy topologies

In this note we lay down criteria for the ‘goodness’ (following Lowen) of definitions of induced and coinduced fuzzy topologies. Both are known (under different terminology) to be good. In the case of induction we give an alternative proof to that of Lowen, using basic open sets in fuzzy topological spaces, thus providing a simple fuzzy topological proof of the extension theorem for lower semi-continuous functions.

Fuzzy sets and functions on fuzzy spaces

We generally follow the terminology of Azad ( J. Math. Anal. Appl., 82 1981, 14–32) and Ming ( J. Math. Anal. Appl., 76 1980, 571–599). In addition to the fundamental concepts for fuzzy sets, we emphasize the usefulness of the concepts of fuzzy point-fuzzy elementhood. For fuzzy sets A and B a new characterization is given for relations A ⊂ B and A = B. This knowledge permits us to combine the two definitions of fuzzy point-fuzzy elementhood. In the third section some results are given concerning various special types of fuzzy sets in fuzzy topological spaces, and the fuzzy semi-regular, fuzzy regular spaces defined by Azad. In the last section, the definitions of H. almost continuous, W. almost open functions Urysohn space which are defined by Hussain, Wilansky, and Noiri, respectively, are extended to fuzzy sets. Furthermore some results are obtained in the functions of the fuzzy topological spaces defined by Azad and those are defined here.

Discrete sets in fuzzy topological spaces

We introduce the notion of discrete set in fuzzy topological spaces and prove that a fuzzy Hausdorif topological space X has an infinite discrete set if the set X is infinite.

A note on semi-open sets in fuzzy topological spaces

Concepts of semi-T i ( i = 0, 1, 2) spaces and semi-R i ( i = 0, 1) spaces are introduced and some fuzzy topological properties are investigated under the above mentioned axioms. Some mappings involving semi-open sets on fuzzy topological spaces are also introduced and investigated.