Fuzzy weakly completely continuous functions

A Note on Strongly Lower Semi-Continuous Functions and the Induced Fuzzy Topological Space Generated by Them

“…Let x 0 I and  > 0 be arbitrary. Then for all In a similar way, using Theorem 2.7, it can be proved that the Theorem 3.2 in [5] is not true and the modified form of the theorem is given by: It can be inferred from Mack [21] that the inverse image of a regular F  -subset under a continuous mapping is again a regular F  -subset. Now we define two weaker forms of continuous mapping which will be used in the sequel.…”

“…Theorem 1 of[5], follows as a corollary of Theorem 2.6. That Theorem 1 of[5] is not valid can be seen in Example 3.11.…”

“…In [5], Bhaumik and Mukherjee introduced completely induced fuzzy topological space with the help of completely LSC (CLSC) functions and Mukherjee and Halder [6] introduced -induced fuzzy topological space with the help of -LSC functions, and various properties of the respective spaces were studied. It was observed that the various properties of the completely induced fuzzy topological space studied in [7,8] are based on Theorem 3.2 of [5], which states that "a fuzzy subset A in completely induced fuzzy topological space (X, C(T)) is open iff for each rI, the strong r-cut  r (A) is regular open in the topological space (X, T)". But, incidentally, it has been noticed that the above result is not always valid.…”

“…In fact, the modified version of the theorem would be "a fuzzy subset A in completely induced fuzzy topological space (X, C(T)) is open iff for each rI, the strong r-cut  r (A) is a union of regular open sets in the topological space (X, T)", which is shown in Theorem 3.7 of this paper. Actually, Theorem 3.2 of [5] is a consequence of Theorem 1 of [5] which states that "the necessary and sufficient condition for a real valued function f to be CLSC is that for all aR, the set {xX: f(x) > a} is regular open (or equivalently {xX: f(x)  a} is regular closed)". That this is not true can be seen in Note 3.11 of this paper.…”

On Fuzzy Weakly Semiopen Functions

“…This condition when combined with fuzzy weak semiopenness imply fuzzy semiopenness. Recall that, a function f : (X, τ 1 ) → (Y, τ 2 ) is said to be fuzzy strongly continuous [2], if for every fuzzy subset λ of X, f (Cl(λ)) ≤ f (λ).…”