Fuzzy Real Line Research Articles

The order aspect of the fuzzy real line

It is the purpose of this paper to put the “fuzzy real line” in a setting which proves to be advantageous to a more fundamental study of that space. Actually there are three different fuzzy real lines to be found in the literature, mainly defined by U. Hohle in [1], [2] and by B. Hutton in [4], [5], and a fourth one shall be added in this work. The main result of this paper is the fact that three of the four spaces are homeomorphic to fuzzy topological spaces the underlying sets of which are, in each case, the probability measures on ℝ, and the fuzzy (resp. quasi fuzzy and translation-closed fuzzy) topologies of which are determined by the left and right sections of a canonical fuzzy extension of the strict order relation on ℝ. From this it will follow very fundamentally that it is the order of ℝ, and not the topology, which determines the fuzzy real line.

Complete fuzzy topological hyperfields and fuzzy multiplication in the fuzzy real lines

We construct a fuzzy multiplication for the case L a chain in the L-fuzzy real line R ( L) which is jointly fuzzy continuous and which is the unique, piecewise consistent extension of ordinary multiplication to R ( L) distributing over the fuzzy addition constructed by the author in [47]; we also specify the fuzzy integers and give an extensive analysis of fuzzy contractions and expansions. These results, the constructions of this paper, and the results of [47] are used to prove R ( L) and the stratified R c( L) are nontopologically isomorphic complete fuzzy topological hyperfields if L is a chain with | L|⩾3. We can now completely answer for L a chain the following question of [27] in the affirmative: can a set of fuzzy numbers, its fuzzy topology, and its fuzzy addition and multiplication be defined such that these operations are jointly fuzzy continuous extensions of the usual addition and multiplication of R and can we specify the fuzzy integers? These results, in combination with recent papers of several authors, would seem to answer in the affirmative the following philosophical question (cf. [34]): does fuzzy topology have deep, specific, canonical examples?

Fuzzy integers, fuzzy rationals and other subspaces of the fuzzy real line

It is the purpose of this note to draw the attention to some noteworthy subspaces of the fuzzy real line M <( R ) which can be characterized by means of classical well known sequence spaces and function spaces.

Compactness properties in the fuzzy real line

It is the purpose of this paper to give the solution to a number of open questions in [14], concerning the compactness properties of the fuzzy real line and the fuzzy real unit interval. However using the model of the fuzzy real line which we already proposed in [9, 12] we are able to describe all compactness properties not only of the spaces mentioned above but actually of several large classes of subspaces of the fuzzy real line; and thus the answers to the questions in [14] follow as easy consequences of our general results.

Separation axioms and the fuzzy real lines

We show that for each lattice L which is completely distributive and equipped with an order-reversing involution, the canonical topology of the fuzzy real line ?(L) is uniformizable in the sense of Hutton; furthermore, this uniformity induces a pseudometric which induces the canonical topology. Consequently, ?(L) satisfies the higher order separation axioms of Hutton and Sarkar. More generally, we answer most of the open questions concerning ?(L) and its subspaces with respect to each of the separation axiom schemes of Hutton and Sarkar, answer some of these same questions for the stratified fuzzy real line ?c(L) and its subspaces, and analyze ?c(L) and its subspaces with respect to the separation axioms of Rodabaugh.

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.