This paper considers fuzzifying topologies, a special case of I-fuzzy topologies (bifuzzy topologies), introduced by Ying [25]. It investigates topological notions defined by means of  -open sets when these are planted into the framework of Ying’s fuzzifying topological spaces (by Łukasiewicz logic in [0, 1]). In this paper we introduce some sorts of operations, called general fuzzifying opera...

In this paper, the theory of γ -convergence and γ c -convergence on nets and filters is established in fuzzifying topology. Some important and interesting results in fuzzifying topology are obtained by means of the theory.

The concepts of fuzzy e-open sets and fuzzy e-continuous are introduced and studied in fuzzifying topology and by making use of these concepts, we introduce and study T e 0 -, R e 0-, T e 1 -, R e 1-, T e 2 -, T e 3 -, T e 4 -, strong T e 3 and strong T e 4 separation axioms in fuzzifying topology and give some of their characterizations as well as the relations of these axioms and other separa...

For a matroid with an ordered (or “labelled”) basis, a basis exchange step removes one element with label l and replaces it by a new element that results in a new basis, and with the new element assigned label l. We prove that one labelled basis can be reconfigured to another if and only if for every label, the initial and final elements with that label lie in the same connected component of th...

Matroid theory gives us powerful techniques for understanding combinatorial optimization problems and for designing polynomialtime algorithms. However, several natural matroid problems, such as 3-matroid intersection, are NP-hard. Here we investigate these problems from the parameterized complexity point of view: instead of the trivial O(n) time brute force algorithm for finding a k-element sol...

In this paper, some characterizations of fuzzifying strong compactness are given, including characterizations in terms of nets and pre -subbases. Several characterizations of locally strong compactness in the framework of fuzzifying topology are introduced and the mapping theorems are obtained.

A matroid M is said to be k–connected up to separators of size l if whenever A is (k − 1)–separating in M , then either |A| ≤ l or |E(M)− A| ≤ l. We use si(M) and co(M) to denote the simplification and cosimplification of the matroid M . We prove that if a 3–connected matroid M is 4–connected up to separators of size 5, then there is an element x of M such that either co(M\x) or si(M/x) is 3–co...

This paper presents characterizations of M-fuzzifying matroids bymeans of two kinds of fuzzy operators, called M-fuzzifying derived operatorsand M-fuzzifying difference derived operators.

Blocks of a matroid are called hyperplanes. For various definitions and results connected with matroids, see [26]. Subsets of X , which are intersections of hyperplanes are called flats of a matroid. Each subset Y c_ X has a well-defined rank. If F is a flat of rank i and x e X \ F , then, there is a unique flat of rank (i + 1) which contains FU{x}. Rank of X is said to be the rank of matroid. ...