Inverse system and Inverse Limit of Intuitionistic Fuzzy Topological Spaces
by
Cigdem Gunduz (Aras)
Department of Mathematics, Kocaeli University, 41380,Kocaeli Turkey

In the talk, we discuss inverse systems and inverse limits in the category of intuitionistic fuzzy topological spaces and establish some of their properties. Let IFTS be the category of intuitionistic fuzzy topological spaces and J be direct poset (consider as a category).

Definition 1. Any functor D:J^op → IFTS is called an inverse system in IFTS, the limit of D is called an inverse limit of D.

Theorem 2. Every inverse system in the category of IFTS has a limit, and this limit is unique.

Theorem 3. Let Inv(IFTS) be a category of all inverse systems in IFTS and all mappings between them. Then lim operation is a functor from the category of Inv(IFTS) to the category of IFTS.

Lemma 4. Let f:( I^X, t ) → ( I^Y, s) be a mapping of IFTSs.

fis an intuitionistic fuzzy open (closed) gp-map if and only if f:(I^X, t^r ) → ( I^Y, s^r ) is a fuzzy open (closed) for each r ∈ I₀ .

Theorem 5. Let f = ( j: J'→J,  { f_i' :I^{X_{j( i' )}} → I^Y_i' }_{i' ∈ J'} ) be a morphism from the inverse system X = {I^X_i }_{i ∈ J} to the inverse system Y = { I^Y_i' }_{i' ∈ J'} in the category of Inv(IFTS). If f_i' is injective (bijective) gp-map for each i' ∈ J', then limf:limX → limY is injective (bijective) gp-map.

Theorem 6. If ({(I^X_i, t_i)}_{i ∈ J}, {p_i^i'}_{i < i'} is an inverse system of fuzzy compact Hausdorff spaces, then limI^X_i is fuzzy compact space.

References

[1] T.K. Mondal, S.K.Samanta, On intuitionistic gradation of openness, Fuzzy Sets and Systems 131 , 323-336 (2002)

[2] S. -G. Li, Inverse limits in category LTop( I)¹, Fuzzy Sets and Systems 108 , 235- 241(1999)

Date received: March 28, 2008