Finitistic Spaces in L-topological Spaces

In this paper the concept of finitistic spaces in L-topological spaces is introduced by means of α-Q-covers of open L-subsets. Also a characterization of finitistic spaces in the weakly induced L-topological spaces is obtained. Moreover the behavior of fini-tistic spaces under various types of maps such as fuzzy perfect maps is also investigated.


Introduction
Fuzzy mathematics begins with Zadeh's paper of 1965, has come of age.The language and behavior of it have developed accordingly.The concept of finitistic spaces in general topology was introduced by R. G. Swan [16] in 1960.But the term finitistic was used by Bredon [2].Many results related to finitistic spaces also known as boundedly metacompact spaces can be seen in [4,5,6,7,9,10].In [12], Jamwal and Shakeel introduced the concept of finitistic spaces in fuzzy topological spaces by using the usual open cover in L-fuzzy topology.In [1], authors have introduced the concept of covering dimension in L-topological spaces using Quasi-coincidence relation and α-Q-covers and obtained a characterization for it.Motivated by this definition of local covering dimension in L-topological spaces, in this paper we introduce the concept of finitistic spaces in L-topological spaces using α-Q-covers and will study its various properties.Besides getting a characterization for finitistic spaces in the weakly induced L-topological spaces, it is also shown that every fuzzy paracompact space with finite covering dimension is finitistic and finitistic property is preserved by fuzzy perfect maps.
A molecule or co-prime element in a lattice L is a join irreducible element in L and the set of all non zero co-prime elements of L is denoted by M (L).Also we denote A (α) = {x ∈ X : A(x) 6 ≤ α} and A complete lattice L is completely distributive if it satisfies either of the logically equivalent CD1 or CD2 below: CD1: If (L, 0 ) is a complete lattice, then for a set X, L X is the complete lattice of all maps from X into L, called L-sets or L-subsets of X.Under point-wise ordering, a ≤ b in L X if and only if a(x) ≤ b(x) in L for all x ∈ X.If A ⊂ X, 1 A ∈ 2 X ⊂ L X is the characteristic function of A. The constant member of L X with value α is denoted by α itself.We use the same notation to represent crisp set as well as its characteristic function.Wang [17] proved that a complete lattice is completely distributive if and only if for each α ∈ L, there exists B ⊆ L such that (i) a = ∨A and (ii) if A ⊆ L and a ≤ ∨B, then for each b ∈ B, there exists c ∈ A such that b ≤ c.B is called the minimal set of a and β(a) denote the union of all minimal sets of a. Again β * (a) = β(a) ∩ M (L).Clearly β(a) and β * (a) are minimal sets of a.
Clearly L X has a quasi complementation 0 defined point-wisely α 0 (x) = α(x) 0 for all α ∈ L and x ∈ X.Thus the DeMorgan laws are inherited by (L X , 0 ).
Let (L, 0 ) be a complete lattice equipped with an order reversing involution and X be any non empty set.A subfamily τ ⊂ L X which is closed under the formation of sups and finite infs (both formed in L X ) is called an L-topology on X and its members are called open L-sets.The pair (X, τ ) is called an L-topological space (L − ts).The category of all L-topological spaces, together with L-continuous mappings and the composition and identities of Set is denoted by L-Top.Quasi complements of open L-sets are called closed L-sets.
We know that the set of all non zero co-prime elements in a completely distributive lattice is ∨-generating.Moreover for a continuous lattice L and a topological space (X, T ), T = i L ω L (T ) is not true in general.By proposition 3.5 in Kubiak [13] we know that one sufficient condition for In [18] Wang extended the Lowen functor ω for completely distributive lattices as follows: For a topological space (X, T ), (X, ω(T )) is called the induced space of (X, T ) where In 1992 Kubiak also extended the Lowen functor ω L for a complete lattice L. In fact when Based on these facts, in this paper we use a complete, completely distributive lattice L in L X .For a standardized basic fixed-basis terminology, we follow Hohle and Rodabaugh [11].Also L-Pnt (X) denote the collection of all L-fuzzy points in the L-ts (X, τ ).Definition 2.1.[19] Let (X, τ ), (Y, µ) be L-topological spaces, f : X → Y be an ordinary mapping.Based on this we define the L-fuzzy mapping f → : L X → L Y and its L-fuzzy reverse mapping Definition 2.7.[1] Let U = {U λ : λ ∈ Λ}, not all zero, be a family of L-subsets of an L-ts X.The order of a fuzzy point x α in U is the number of elements of U which are quasi coincident with x α .We denote it by Ord (x α , U).The order of a collection U is defined as the largest integer n such that for every x α with α ∈ M (L), x α quasi coincides with (n + 1) members of U. i.e., Ord (x α , U) ≥ n + 1 for all α ∈ M (L).

Definition 2.8. A collection A refines a collection B (A < B) if for every
A ∈ A, there exists B ∈ B such that A ≤ B.

Now by the weakly induced property,
W is an open cover of X.Now we will show that W also has an order not exceeding n.For, if possible, let order of W be greater than n.Therefore there exist x ∈ X which belongs to at least n + 2 members of W. i.e., x ∈ {x ∈ X : V (x) 6 ≤ α 0 } for at least n + 2 members of V i.e., V (x) 6 ≤ α 0 for at least n + 2 members of V or x α ≺ V for at least n + 2 members of V.This is contradiction to that order of V is not exceeding n.Hence (X, [τ ]) is finitistic.(iii) ⇒ (i): Suppose (X, [τ ]) is finitistic.We have to show that (X, τ ) is finitistic.
Let U ⊂ τ be an open α-Q-cover of > where α ∈ M (L).Since (X, τ ) is weakly induced it follows that {U (α 0 ) : U ∈ U} is an open cover of X and it has an open refinement V of some finite order say n.For every V ∈ V let U V be such that V < U V (α 0 ) .Consider W = {χ V ∧ U V : V ∈ V, V < U V (α 0 ) }.This is an open refinement of U with order n.Hence (X, τ ) is finitistic. 2 Theorem 3.4.Let (X, τ ) be a finitely supported L-ts.Then (X, τ ) is finitistic.

Definition 3 . 1 .Theorem 3 . 2 . 2 Theorem 3 . 3 .
Let (X, τ ) be an L-ts, D ∈ L X .D is said to be α-finitistic if every α-Q-cover of D has an α-Q-cover refinement by open L-sets with finite order.D is finitistic if D is α-finitistic for every α ∈ M (L).And (X, τ ) is finitistic if > is finitistic.Every closed subspace of a finitistic space is finitistic.Proof.Let (X, τ ) be a finitistic space.And Y be a closed subspace ofX.Then clearly Y 0 is an open L-set in X.Let A = {A λ : λ ∈ Λ} be an open α-Q-cover of Y .Then for every A λ ∈ A there exist open L-subsets B λ of X such that A λ = B λ ∧ Y .Now consider B = {B λ : A λ = B λ ∧ Y ∀A λ ∈ A} ∪ Y 0 .Then clearly B is an open α-Q-cover of X by open L-sets.Since (X,τ ) is finitistic, for every α ∈ M (L), B has a finite order α-Q-cover refinement say C by open L-sets.Then D = {C ∧ Y : C ∈ C} is a finite order α-Q-cover refinement of A by open L-sets and hence Y is finitistic.In a weakly induced L-ts the following are equivalent: