Locally Starplus-Compactness in L-Topological Spaces

ABSTRACT The notion of local starplus-compactness on an L-fuzzy topological space, which is an extension of the notion of local compactness in general topology, is introduced. It turns out that local starplus-compactness is finitely productive, closed hereditary and invariant under fuzzy continuous open surjections. Moreover, local starplus-compactness is a good extension of the notion of local compactness in general topology. Examples are included to show that local starplus-compactness is neither hereditary nor expansive, nor contractive.


Introduction
The class of locally compact spaces is far more wider than the class of compact spaces. The locally compact spaces often arise in topology and applications of topology to geometry, analysis and algebra. For example, the study of locally compact abelian group forms the foundation of harmonic analysis. It is well known that every compact space is locally compact but the converse need not be true. For example, the Euclidean space R is locally compact but not compact. Topological manifolds share the local properties of Euclidean space and hence are locally compact. A locally compact space can be imbedded in a compact space, which is its compactification. One of the simplest compapctification of a space is the one point compactification, wherein one simply adjoins one new point to the space. The classical example of one point compactification is the embedding of the Gaussian plane of complex numbers into the Riemann sphere. The category of locally compact spaces has been applied in almost every subdiscipline of mathematics and hence it is important to formulate an appropriate version of local compactness in the L-fuzzy setting.
The notion of compactness in fuzzy topology have been thoroughly investigated by various authors (see [1][2][3][4][5][6][7][8][9][10][11][12][13]). However, a satisfactory theory for the localisation of the notion of compactness in fuzzy topology has not been established because of the absence of proper definition of subspace on an arbitrary fuzzy subset. Kudri and Warner [11] defined a notion of L-fuzzy local compactness on an L-fuzzy topology by using very compact neighbourhood instead of compact neighbourhood of a fuzzy point. In [9] Kohli and Prasannan introduced the notion of starplus-compactness for a fuzzy topological space and successfully applied it to study fuzzy topologies and fuzzy uniformities on function spaces [12,14].
In this paper we localise the notion of starplus-compactness on an L-topological space, which is an extension of the notion of local compactness to the L-fuzzy setting. It turns out that local starplus-compactness is finitely productive, closed hereditary and invariant under continuous open surjection. Moreover, local starplus-compactness is a good extension of the notion of local compactness in general topology. Examples are included to reflected upon that local starplus-compactness is neither hereditary nor expansive nor contractive.

Basic Definitions and Preliminaries
Throughout this paper X is a nonempty set, (L, ≤, V, , ) is a complete De Morgan algebra with the smallest element and the largest element are denoted by 0 and 1, respectively. An L-fuzzy subset on X is a mapping A : X → L. The symbol L X will denote the set of all Lfuzzy sets (or L-sets, for short) on X. The smallest element and the largest element in L X are denoted by 0 and 1, respectively. A crisp subset A of X is denoted by its characteristic function χ A ∈ L X . The elements of L X will be denoted by the letters A, B, C etc. If c ∈ L, then the constant fuzzy set with value c is denoted by c. The constant L-set taking each member of X into 0 and 1 are denoted by 0 X and 1 X , respectively.
An element a ∈ L is said to be a prime element if a ≥ b ∧ c ⇒ a ≥ b or a ≥ c. The set of all non unit prime elements in L is denoted by P(L). An element a ∈ L is called a co-prime if a ∈ P(L). The set of all non zero co-prime elements in L is denoted by M(L) and set of all non zero co-prime elements in L X is denoted by M(L X ).

Definition 2.1 ([3,12]):
For an L-set A in X, the set A (a) = {x ∈ X : A(x) a, a ∈ P(L)} is called the strong a-level set of A. The set {x ∈ X : A(x) > 0} is called the support of A and is denoted by suppA.

Definition 2.2:
An L-fuzzy point on X is an L-set x a ∈ L X , defined by An L-set A is said to contain a fuzzy point x a if A(x) ≥ a, and it is denoted by x a ≤ A. An L-topological space (or L-space for short) is a pair (X, τ ), where τ ⊆ L X contains 0 X , 1 X and is closed for arbitrary suprima and finite infima. τ is called an L-topology and members of τ are called open L-sets.  If (X, τ ) is an L-topological space then for each a ∈ P(L), the collection i a (τ ) = {A (a) : A ∈ τ } is a topology on X; we shall call it the strong a-level topology. Finally, for an L-topology τ on X, i(τ ) is the topology generated by taking the collection ∪{i a (τ ) : a ∈ P(L)} as a subbase.

Definition 2.5 ([10,17]):
Let (X,T) be a topological space and let ω L (T) denote the collection of all lower semicontinuous functions from X into L equipped with the lower Scott topology, i.e, ω L (T) = {A ∈ L X : A (a) ∈ T for each a ∈ L}. Then ω L (T) is a laminated (also called stratified) L-topology (L-topology which contains all constant L-sets) on X and is called the topologically generated L-topology of T and the fts (X, ω L (T)) is called the topologically generated L-topological space (also called the induced L-topological space).

Definition 2.6 ([10]):
An L-topology τ on X is said to be topologically generated if τ = ω L • i L (τ ). We shall often writeτ instead of ω L • i L (τ ) for a fuzzy topology τ on X.
The category of topological spaces and continuous maps is denoted by TOP. Let L be a complete DeMorgan algebra, then the category of all L-topological spaces as objects and the collection of all L-continuous maps as morphisms form a category which is denoted by L-FTS. Lowen [16] was the first researcher who introduced and studied the functors ω L : TOP →L-FTS and i L : L-FTS → TOP for L = I. Later several authors extended these functors into a more genaral setting by replacing I with L (for more details refer [17]). Rodabaugh [18] defined the functor G Z : For notational convenience, we shall denote it byχ(T).

Definition 2.8 ([13]):
Let (X, τ ) be an L-topology. An L-set A ∈ L X is called a pseudo closed fuzzy set with respect to the L-topological space (X, τ ) if the strong a-level sets A (a) are closed in i a (τ ) for each a ∈ P(L). The complement of a pseudo closed fuzzy set will be referred to as a pseudo open fuzzy set. The smallest pseudo closed fuzzy set containing A is called the pseudo closure of A and it is denoted byĀ ps .

Local Starplus-Compactness
In this section we introduce the notion of local starplus-compactness, which is an extension of the notion of local compactness to the L-fuzzy setting. We extend the notion of starpluscompactness defined in [9] to the L-fuzzy setting as follows: compact in (X, i a (τ )) for each a ∈ P(L). The L-topological space (X, τ ) is said to be starplus-compact if (X, i a (τ )) is compact for each a ∈ P(L).

Definition 3.2:
An L-topological space (X, τ ) is said to be locally starplus-compact if every L-fuzzy point in X has a starplus-compact neighbourhood in X.
Let τ be the L-topology on X generated by the collection δ. Then, The indiscrete topology, if α 1 2 , 1 2 .

(i) The L-topological space (X, ω L (T)) is locally starplus-compact if and only if (X, T) is locally compact. (ii) The L-topological space (X, χ(T)) is locally starplus-compact if and only if (X, T) is locally compact.
Proof: (i) Suppose (X, ω L (T)) is locally starplus-compact. Since i a (ω L (T)) = T by Proposition 3.2, it follows that the space (X, T) is locally compact. Conversely, let (X, T) be locally compact and let x a be an L-fuzzy point in X. Then there exists a compact neighbourhood C of x in (X, T). So χ C is a starplus-compact neighbourhood of x a in (X, ω L (T)).
(ii) Proof of (ii) is similar to that of part (i). Hence, the theorem is proved.

Remark 3.2:
The above theorem shows that the notion of local starplus-compactness is a good extension.

Remark 3.3:
Every starplus-compact L-topological space is locally starplus-compact. However, the converse is not true as is exhibited by the following example.

Example 3.2:
Let R be the set of real numbers with the Euclidean topology T. Since, (R, T) is locally compact, the L-topological space (R, ω L (T)) is locally starplus-compact. However, (R, ω L (T)) is not starplus-compact.

Proposition 3.3: A Hausdorff L-topological space is locally starplus-compact if and only if every neighbourhood of each fuzzy point contains a neighbourhood whose pseudo closure is starplus-compact.
Proof: Let X be a Hausdorff starplus-compact L-topological space and let K be a starpluscompact neighbourhood of an L-fuzzy point x a . If U is any neighbourhood of x a , then for each a ∈ P(L), is Hausdorff for each a ∈ P(L). Hence ((U ∧ K) (a) ) ps ⊂ K. Proof of the converse is obvious.

Proposition 3.4: A closed crisp L-subspace of a locally starplus-compact space is locally starplus-compact.
Proof: Let F ⊂ X be a crisp subset of X such that χ F is a closed L-set in (X, τ ). Let x a be an L-fuzzy point in (F, τ F ). Since X is locally starplus-compact, there exist a starplus-compact neighbourhood K of x a in X. Now, since χ F is closed in (X, τ ), F is closed in (X, i a (τ )) for each a ∈ P(L) and so, (K ∧ χ F ) (a) = K (a) ∩ F is compact in (F, i a (τ F )) for each a ∈ P(L). Hence K ∧ χ F is a starplus-compact neighbourhood of x a in (F, τ F ).

Remark 3.4:
The above Proposition shows that local starplus-compactness is closed hereditary. However, it is not hereditary as is reflected in the following example.

Example 3.3:
Let R be the set of real numbers with the Euclidean topology T and Q be the set of rational numbers with the subspace topology T Q . Then (R, T) is locally compact while (Q, T Q ) is not. Consider the L-topology (R, ω L (T)), which is locally starplus-compact. We shall show that the L-subspace (Q, τ Q ) is not locally starplus-compact. Now, U ∈ τ Q if and only if there exist a V ∈ ω L (T) such that U = V ∧ 1 Q . Hence, for each a ∈ P(L) Thus i a (τ Q ) = T Q , for each a ∈ P(L) and (Q, T Q ) is not locally compact. Hence (Q, τ Q ) is not locally starplus-compact.

Proposition 3.5: A finite product of locally starplus-compact L-topological spaces is locally starplus-compact.
Proof: Let {(X j , τ j ) : j = 1, 2, . . . , n} be a finite collection of locally starplus-compact L-topological spaces and let x a = (x 1 , x 2 , . . . , x n ) a be a fuzzy point in the product Lopological space n j=1 X j , n j=1 τ j . Then (x j ) a is a fuzzy point in the L-topological space X j and since X j is locally starplus-compact, there exist a starplus-compact neighbourhood K j , j = 1, 2, .., n of (x j ) a . By Theorem 2.2, n j=1 K j is a starplus-compact neighbourhood of x a . Hence n j=1 X j is locally starplus-compact.

Theorem 3.2:
Let {(X j , τ j ) : j ∈ J} be a collection of laminated L-topological spaces. The product L-topological space j∈J X j is locally starplus-compact if and only if each X j is starpluscompact except for finitely many, which are locally starplus-compact.
Proof: Suppose X = j∈J X j is locally starplus-compact. Since each X j is a laminated L-topological space, then for each j ∈ J, the projection map π j : X → X j is fuzzy continuous open surjection. So by Proposition 3.5, X j is locally starplus-compact. Again, for each L-fuzzy point x a ∈ X, there exist a starplus-compact neighbourhood K of x a . Then each π j (K) is starplus-compact and since π j (K) = X j for all j except for finitely many, the result follows.
Conversely, suppose that all the X j 's are starplus-compact except for a finitely many, which are locally starplus-compact, say X j 1 , X j 2 , . . . , X j n . Then Y = j∈J,j =j i X j is starpluscompact by Theorem 2.1, and so it is locally starplus-compact. Now, the product n i=1 X j i is locally starplus-compact by Proposition 3.6, and hence Y × n i=1 X j i is locally starpluscompact by Proposition 3.6 and the same is homeomorphic to the product L-topological space X.

Remark 3.6:
The following example shows that local starplus-compactness is neither expansive nor contractive.

Example 3.5:
Let X = R n , n ≥ 1 with the Euclidean topology T. Consider the fuzzy topologies ω L (T), τ and τ 0 on X, where τ is the fuzzy topology defined on X in Example 3.1 and τ 0 be the indiscrete fuzzy topology. Then τ 0 ⊂ τ ⊂ ω(T) and the L-topological space (X, ω L (T)) and (X, τ 0 ) are locally starplus-compact whereas the L-topological space (X, τ ) is not. This shows that the notion of locally starplus-compactness is neither expansive nor contractive.

Conclusion
The notion of local starplus-compactness is introduced, which is the extension of the notion of local compactness in general topology to the L-fuzzy setting. On studying its basic properties, it turns out that the category of local starplus-compact fuzzy topological spaces is finitely productive, closed hereditary and that local starplus-compactness is a good extension of the notion of local compactness in general topology. It is shown that the class of locally starplus-compact fuzzy topologies is invariant under L-continuous open surjections.

Disclosure statement
No potential conflict of interest was reported by the author.