Fuzzy Unit Research Articles

On fuzzy metrizability

In this paper we answer two questions raised by M. A. Erceg [ J. Math. Anal. Appl . 69 (1979), 205–230]: precisely we show that the fuzzy unit interval is never T 0 except in the standard case and that a fuzzy pseudo-metrizable T 0 space is metrizable in the sense of Erceg ( ibid. ); hence the two definitions of metrizable space given in B. Hutton and I. Reilly [ Fuzzy Sets and Systems 3 (1980), 93–104] and Erceg ( ibid. ) are equivalent.

Generating fuzzy topologies with semi-closure operators

Let X be an L-fuzzy topological space. For α ε L−1 and A ⊂ X, x εcα(A) if and only if G fuzzy-open and G(x) > α imply there is a ε A with G(a) > 0. Under certain restrictions on α, which are always satisfied if L is linearly ordered, cα is a semi-closure operator. This paper contains necessary and sufficient conditions under which a collection of semi-closure operators generates a fuzzy topology. In general, such a collection does not generate a unique topology. The class of topologies generated is shown to be closed under suprema and its largest member is characterized. The topology for the fuzzy unit interval is characterized among those generated by its α-closure operators.

Suitability in fuzzy topological spaces

An L-fuzzy topological space is said to be suitable if it possesses a nontrivial crisp closed subset. Basic properties of and sufficient conditions for suitable spaces are derived. Characterizations of the suitable subspaces of the fuzzy unit interval, the fuzzy open unit interval, and the fuzzy real line are obtained. Suitability is L-fuzzy productive; nondegenerate 1 ∗-Hausdorff spaces are suitable; the fuzzy unit interval, the fuzzy open unit interval, and the fuzzy real line are not suitable; and no suitable subspace of the fuzzy unit interval, the fuzzy open unit interval, or the fuzzy real line is a fuzzy retract of the fuzzy unit interval, the fuzzy open unit interval, or the fuzzy real line, respectively. Without restrictions there cannot be a fuzzy extension theorem.

The Hausdorff separation axiom for fuzzy topological spaces

We develop a theory of α-Hausdorff fuzzy topological spaces which is compatible with α-compactness and fuzzy continuity, and for α a certain type of member of a given lattice we obtain characterizations of the α-Hausdorff subspaces of the fuzzy unit interval, the fuzzy open unit interval, and the fuzzy real line. In route we give an easy proof of the Fuzzy Tychonov Theorem for α-compactness and extend the theory of one-point α-compactifications.

Compactness in fuzzy topological spaces

Since earlier approaches to compactness in fuzzy spaces have serious limitations, we propose a new definition of fuzzy space compactness. In doing so, we observe that it is possible to have degrees of compactness, which we call α-compactness (α a member of a designated lattice). We obtain a Tychonoff Theorem for an arbitrary product of α-compact fuzzy spaces and a 1-point compactification. We prove that the fuzzy unit interval is α-compact.

Uniformities on fuzzy topological spaces

In this paper we extend the notions of quasi-uniformities on topological spaces to fuzzy topological space. We prove theorems corresponding to many of the usual theorems. In particular we show every fuzzy topological space is quasi-uniformizable and characterize uniformizability in terms of a type of complete regularity. We also construct a natural uniformity on the fuzzy unit interval.

A Fuzzy Recurrent Neural Network Of BinaryNeurons For Content Addressable Memory

This paper is concerned with a proposal for a recurrent neural network of fuzzy neurons which may be used as a content addressable memory. The behavior of the fuzzy unit in the network is based on fuzzy logic in that each component of the binary input vector to the fuzzy neuron is compared to a number which represents the membership value for a 0 in that position. The results of the comparisons are then combined using a generalized mean function to produce a single number which is compared to a threshold. A training algorithm is developed based on an algorithm for linear inequalities described by Ho and Kashyap in a paper titled “ An Algorithm for Linear Inequalities and its Applications”. The results obtained by simulation of this content addressable memory look promising enough to warrant further investigation.