Compactness in fuzzy topological spaces

Ulam-Hyers stability of a 2-variable AC-mixed type functional equation in Felbin's type spaces: fixed point method

“…In [12] Grantner takes the fuzzy real number as a decreasing mapping from the real line to the unit interval or lattice in general. Lowen [23] applies the fuzzy real numbers as non-decreasing, left continuous mapping from the real line to the unit interval so that its supremum over R is 1.…”

Gradual Numbers and Their Application to Fuzzy Interval Analysis

“…For fuzzy real numbers, the transition from one side of ¥ to the other is made gradual. Often, this notion takes the form of a decreasing mapping from the reals to the unit interval or a suitable lattice (Grantner et al [28]), or a probability distribution function (Lowen [19]); variants of such a fuzzy reals were also studied by Rodabaugh [17] and Höhle [18]. The sum of two fuzzy Hutton real numbers …”

Filter Convergence and Fuzzy Topology

“…[13] showed that a fuzzy topological space (X, δ) is ultrafilter α-compact for Convergence (VI) iff it is strong fuzzy compact. In [2], the Tychonoff theorems for α-compactness and strong compactness, respectively, were proved using the Alexander Subbase Theorem. Therefore our approach using ultrafilter provides a different and simple proof for the Tychonoff theorem.…”

“…Moreover it is interesting to know that there exist various notions of convergence in a fuzzy topology [1,3,4,5,8,9,14,18]. On the other hand there exist various notions of compactness in a fuzzy topology using open sets, prefilters, fuzzy nets and functors between fuzzy topological spaces and topological spaces [2,6,7,10,11,15,17]. This means that we can discuss various types of compactness in fuzzy topology in terms of convergence.…”

“…After finding an universal method to define the notions of compactness using cluster points and convergence of a prefilter, we prove the Tychonoff theorem using characterizations of ultra(maximal) prefilters. In [2], the Tychonoff theorems for α -compactness and strong compactness, respectively, were proved using the Alexander Subbase Theorem. It is interesting to note that our approach provides a simple proof for the Tychonoff theorems as good extensions of compactness in a topological space.…”