On the Level Spaces of Fuzzy Topological Spaces ∗
S. S. Benchalli & G.P. Siddapur
Abstract
It is known that if (𝑋, 𝑇) is a fuzzy topological space and 0 ≤ 𝛼 < 1 then the family𝑇𝛼={𝛼(𝐺) :𝐺∈𝑇}where𝛼(𝐺) ={𝑥∈𝑋:𝐺(𝑥)> 𝛼}, forms a topology on𝑋. In the present paper some level properties have been modified and it is proved that a fuzzy topological space (𝑋, 𝑇) is 𝛼-compact (resp. 𝛼-Hausdorff, countably 𝛼-compact, 𝛼-Lindel¨𝑜f, 𝛼- connected, locally 𝛼-compact) if and only if the corresponding 𝛼-level topological space (𝑋, 𝑇𝛼) is compact (resp. Hausdorff, countably com- pact, Lindel¨𝑜f, connected, locally compact). Some basic properties of𝛼- level sets have also been obtained.
1 Introduction
The investigation of fuzzy topological spaces by considering the properties which a space may have to a certain degree or level was initiated by Gantner et. al [3]. This approach resulted into the investigation of 𝛼-Hausdorff axiom [10], countable 𝛼-compactness,𝛼-Lindel¨𝑜f property [6], local 𝛼-compactness [7], 𝛼- closure [4] etc. in fuzzy topological spaces.
Throughout this paper Chang’s [1] definition of fuzzy topological space (ab- breviated as fts) is used. If 𝑋 is a set and𝑇 is a family of fuzzy subsets of𝑋 satisfying the following conditions (i) to (iii) then T is called a fuzzy topology on𝑋 ; (i)𝑋, 𝜙∈𝑇 (ii) arbitrary union of members of𝑇 is again a member of 𝑇 and (iii) intersection of finitely many members of𝑇 is again a member of𝑇. Further (𝑋, 𝑇) is called a fuzzy topological space (fts). If (𝑋, 𝑇) is a fts and 0≤𝛼 <1 then the family𝑇𝛼={𝛼(𝐺) :𝐺∈𝑇}, of all subsets of𝑋 of the form 𝛼(𝐺) ={𝑥∈𝑋:𝐺(𝑥)> 𝛼} called𝛼-level sets, forms a topology on𝑋 [4] and is called the𝛼-level topology on𝑋.
In this paper, some basic properties of𝛼-level sets have been obtained. The 𝛼-Hausdorff axiom [10] and the local 𝛼-compactness of [7] have been modified.
The 𝛼-connectedness has been proposed. It is proved that a fts (𝑋, 𝑇) is 𝛼- compact (𝛼-Hausdorff, countably 𝛼-compact, 𝛼-Lindel¨𝑜f, 𝛼-connected, locally 𝛼-compact) if and only if the corresponding 𝛼-level topological space (𝑋, 𝑇𝛼) is compact (resp. Hausdorff, countably compact, Lindel¨𝑜f, connected, locally compact)
∗Mathematics Subject Classifications 2000: 54A40, 54D05, 54D20, 54D30, 54D45.2.
Key words: Fuzzy topological spaces, 𝛼-level sets, 𝛼-Hausdorff,𝛼-compact, locally 𝛼- compact,𝛼-Lindel¨𝑜f,𝛼-connected fts.
c
⃝2009 Universiteti i Prishtin¨es, Prishtin¨e, Kosov¨e.
Submitted May, 2009. Published September, 2009.
57
2 𝛼-Level Sets and Their Basic Properties
If 𝐺 is any fuzzy set in a set 𝑋 and 0 ≤ 𝛼 < 1 (0 < 𝛼 ≤ 1) then 𝛼(𝐺) = {𝑥∈𝑋 :𝐺(𝑥)> 𝛼} (resp. 𝛼∗(𝐺) = {𝑥∈𝑋 :𝐺(𝑥)≥𝛼}) is called an 𝛼-level (resp. 𝛼∗-level) set in𝑋.
The term crisp subset refers to an ordinary subset which is identified with its characteristic function as a fuzzy subset.
If𝑓 :𝑋 →𝑌 is a function and𝐴is a fuzzy subset of𝑋 then𝑓(𝐴) is a fuzzy subset of 𝑌 defined by 𝑓(𝐴)(𝑦) = 𝑠𝑢𝑝{
𝐴(𝑥) :𝑥∈𝑓−1(𝑦)}
for each 𝑦 ∈ 𝑌. Further, if𝐵 is a fuzzy subset of Y then𝑓−1(𝐵) is a fuzzy subset of𝑋 defined by𝑓−1(𝐵)(𝑥) =𝐵(𝑓(𝑥)) for each𝑥∈𝑋.
Some basic properties of𝛼-level sets are given in the following.
Theorem 2.1 Let 𝑋, 𝑌 be any two sets and 0 ≤𝛼 <1. The following state- ments are true.
1. If𝐺is any fuzzy set in 𝑋 then𝐺(𝑥)≤𝛼(𝐺)(𝑥)holds for all 𝑥∈𝑋 with 𝐺(𝑥)> 𝛼.
2. If𝐺≤𝐻 then𝛼(𝐺)⊂𝛼(𝐻)for any two fuzzy sets𝐺,𝐻 in𝑋. 3. 𝛼(𝐺) =𝐺if and only if 𝐺is a crisp susbet of𝑋.
4. 𝛼(𝛼(𝐺)) =𝛼(𝐺)for any fuzzy set𝐺in 𝑋.
5. 𝛼(⋁
𝜆
𝐺𝜆) =∪
𝜆
𝛼(𝐺𝜆)for any family{𝐺𝜆:𝜆∈Λ}of fuzzy sets in 𝑋.
6. 𝛼(⋀
𝜆
𝐺𝜆) =∩
𝜆
𝛼(𝐺𝜆)for any family{𝐺𝜆:𝜆∈Λ}of fuzzy sets in 𝑋. 7. If𝑓 :𝑋 →𝑌, then𝑓(𝛼(𝐺)) =𝛼(𝑓(𝐺))for each fuzzy set𝐺in𝑋. 8. If𝑓 :𝑋 →𝑌, then𝑓−1(𝛼(𝐺)) =𝛼(𝑓−1(𝐺))for each fuzzy set𝐺in𝑌. 9. 𝛼(𝐺×𝐻) =𝛼(𝐺)×𝛼(𝐻)for any two fuzzy sets𝐺, 𝐻 in𝑋 where𝐺×𝐻
is a fuzzy set in 𝑋 ×𝑌 given by (𝐺×𝐻)(𝑥, 𝑦) = 𝐺(𝑥)∧𝐻(𝑦) for each (𝑥, 𝑦)∈𝑋×𝑌.
Proof. (1). Let 𝑥∈ 𝑋 with 𝐺(𝑥)> 𝛼. Then 𝑥∈𝛼(𝐺) so that (𝛼(𝐺))(𝑥) = 1≥𝐺(𝑥)> 𝛼 and therefore𝐺(𝑥)≤(𝛼(𝐺))(𝑥).
(2) If 𝑥 ∈ 𝛼(𝐺) then 𝐺(𝑥) > 𝑎 and therefore 𝐻(𝑥) ≥ 𝐺(𝑥) > 𝛼. Conse- quently𝑥∈𝛼(𝐻).
(3) If 𝐺is crisp and if𝑥∈𝑋 then𝐺(𝑥) = 0 or 1. If 𝐺(𝑥) = 0 then𝑥 /∈𝛼(𝐺) and therefore (𝛼(𝐺))(𝑥) = 0 which proves𝐺(𝑥) =𝛼(𝐺(𝑥)). In case if𝐺(𝑥) = 1, then𝐺(𝑥) = 1> 𝛼and therefore𝑥∈𝛼(𝐺) which proves (𝛼(𝐺))(𝑥) = 1 =𝐺(𝑥).
The converse part follows as𝛼(𝐺) is crisp.
(4) Follows from (3) as𝛼(𝐺) is crisp.
(5) If 𝑥 ∈ 𝛼(⋁
𝜆
𝐺𝜆) then 𝑆𝑢𝑝(𝐺𝜆(𝑥)) > 𝛼. Consequently there exists a 𝜆𝑜
such that 𝐺𝜆𝑜(𝑥) > 𝛼 which implies 𝑥 ∈ 𝛼(𝐺𝜆𝑜) and hence 𝑥 ∈ ∪
𝜆
𝛼(𝐺𝜆).
Therefore𝛼(⋁
𝜆
𝐺𝜆)⊂∪
𝜆
𝛼(𝐺𝜆). Similarly∪
𝜆
𝛼(𝐺𝜆)⊂𝛼(⋁
𝜆
𝐺𝜆) and hence the equality.
(6) If 𝑥 ∈ 𝛼(⋀
𝜆
𝐺𝜆) then (⋀
𝜆
𝐺𝜆)(𝑥) > 𝛼 and therefore 𝐺𝜆(𝑥) > 𝛼 for each 𝜆. This implies that 𝑥∈𝛼(𝐺𝜆) for each𝜆and therefore 𝑥∈⋀
𝜆
𝛼(𝐺𝜆). Thus 𝛼(⋀
𝜆
𝐺𝜆)⊂∩
𝜆
𝛼(𝐺𝜆). Similarly∩
𝜆
𝛼(𝐺𝜆)⊂𝛼(⋀
𝜆
𝐺𝜆).
(7) If 𝑦 ∈ 𝑓(𝛼(𝐺)) then there is an element 𝑥 ∈ 𝛼(𝐺) such that 𝑦 = 𝑓(𝑥).
Now 𝐺(𝑥) > 𝛼 and therefore 𝑆𝑢𝑝{
𝐺(𝑥) :𝑥∈𝑓−1(𝑦)}
> 𝛼 which implies (𝑓(𝐺))(𝑦) > 𝛼. Then 𝑦 ∈ 𝛼(𝑓(𝐺)). Thus 𝑓(𝛼(𝐺)) ⊂ 𝛼(𝑓(𝐺)). Similarly it can be shown that 𝛼(𝑓(𝐺))⊂𝑓(𝛼(𝐺)) and hence the result follows.
(8) Let 𝑥∈ 𝑓−1(𝛼(𝐺)). Then 𝑓(𝑥) =𝑦 ∈𝛼(𝐺) so that 𝐺(𝑦) =𝐺(𝑓(𝑥))> 𝛼.
Therefore[
𝑓−1(𝐺)]
(𝑥)> 𝛼which implies𝑥∈𝛼[
𝑓−1(𝐺)]
and hence it follows that 𝑓−1(𝛼(𝐺)) ⊂ 𝛼(𝑓−1(𝐺)). Similarly 𝛼(𝑓−1(𝐺)) ⊂ 𝑓−1(𝛼(𝐺)) and hence the equality.
(9) If (𝑥, 𝑦) ∈ 𝛼(𝐺×𝐻) then (𝐺×𝐻)(𝑥, 𝑦) > 𝛼 and therefore 𝑥 ∈ 𝛼(𝐺) and 𝑦 ∈ 𝛼(𝐻). So (𝑥, 𝑦) ∈ 𝛼(𝐺)×𝛼(𝐻). Thus 𝛼(𝐺×𝐻) ⊂ 𝛼(𝐺)×𝛼(𝐻).
Similarly it can be shown that𝛼(𝐺)×𝛼(𝐻)⊂𝛼(𝐺×𝐻) and hence the equality follows.
3 Level Spaces and Main Results
In the beginning of this section we deal with Rodabaugh’s [10]𝛼-Hausdorff fts.
Definition 3.1 Let 0 ≤ 𝛼 < 1 (0 < 𝛼 ≤ 1). A fts (𝑋, 𝑇) is said to be 𝛼- Hausdorff (resp. 𝛼∗-Hausdorff ) if for each𝑥,𝑦 in𝑋 with𝑥∕=𝑦, there exist 𝐺, 𝐻 in 𝑇 such that𝐺(𝑥)> 𝛼 (resp. 𝐺(𝑥)≥𝛼),𝐻(𝑦)> 𝛼(resp. 𝐻(𝑦)≥𝛼) and 𝐺∧𝐻= 0.
We have the following
Theorem 3.2 Let 0 ≤𝛼 <1. If a fts (𝑋, 𝑇) is 𝛼-Hausdorff, then (𝑋, 𝑇𝛼) is Hausdorff topological space.
Proof. Let𝑥, 𝑦∈𝑋 with𝑥∕=𝑦. Then there are𝐺, 𝐻in𝑇 such that𝐺(𝑥)> 𝛼, 𝐻(𝑦)> 𝛼and 𝐺∧𝐻 = 0. Then𝛼(𝐺) and 𝛼(𝐻) are open sets in (𝑋, 𝑇𝛼) and 𝑥∈𝛼(𝐺),𝑦∈𝛼(𝐻). Also𝛼(𝐺)∩𝛼(𝐻) =𝜙since𝐺∧𝐻 = 0. Hence (𝑋, 𝑇𝛼) is Hausdorff topological space.
The converse of the above theorem holds for the case of 𝛼 = 0, which is given in the following.
Theorem 3.3 Let (𝑋, 𝑇) be a fts. If (𝑋, 𝑇0) is Hausdorff topological space, then(𝑋, 𝑇)is0-Hausdorff fts.
Proof. Let𝑥, 𝑦∈𝑋 with𝑥∕=𝑦. Then there are open sets𝑈, 𝑉 in (𝑋, 𝑇0) such that𝑥∈𝑈,𝑦∈𝑉 and𝑈∩𝑉 =𝜙. Let𝑈 = 0(𝐺), 𝑉 = 0(𝐻) for some𝐺, 𝐻in𝑇.
Then it follows that𝐺(𝑥)>0 and𝐻(𝑦)>0. Further𝐺∧𝐻 = 0 as𝑈∩𝑉 =𝜙.
Hence (𝑋, 𝑇) is 0-Hausdorff.
Definition 3.4 Let𝑋 be a set and0≤𝛼 <1 (0< 𝛼≤1). A family{𝐺𝜆}𝜆 of fuzzy sets in 𝑋 is said to be𝛼-disjoint (resp. 𝛼∗-disjoint) if ⋀
𝜆
𝐺𝜆 ≤𝛼 (resp.
⋀
𝜆
𝐺𝜆< 𝛼).
It is evident that two fuzzy sets 𝐺, 𝐻 in 𝑋 are 𝛼-disjoint (𝛼∗-disjoint) if and only if for each 𝑥 in 𝑋 either 𝐺(𝑥) ≤ 𝛼 (resp. 𝐺(𝑥) < 𝛼) or 𝐻(𝑥) ≤ 𝛼 (resp.𝐻(𝑥)< 𝛼).
Rodabaugh’s definition is suitably modified in the following.
Definition 3.5 Let 0 ≤ 𝛼 < 1 (0 < 𝛼 ≤ 1). A fts (𝑋, 𝑇) is said to be 𝛼- Hausdorff (resp. 𝛼∗-Hausdorff ) if for each𝑥,𝑦 in𝑋 with𝑥∕=𝑦, there exist𝐺, 𝐻 in𝑇 such that𝐺(𝑥)> 𝛼 (resp. 𝐺(𝑥)≥𝛼),𝐻(𝑦)> 𝛼 (resp. 𝐻(𝑦)≥𝛼) and 𝐺, 𝐻 are 𝛼-disjoint (resp. 𝛼∗-disjoint).
For the modified class of𝛼-Hausdorff fuzzy topological spaces we have the fol- lowing.
Theorem 3.6 Let 0 ≤ 𝛼 < 1. A fts (𝑋, 𝑇) is a 𝛼-Hausdorff if and only if (𝑋, 𝑇𝛼) is Hausdorff topological space.
Proof. Let (𝑋, 𝑇) be𝛼-Hausdorff. Let𝑥, 𝑦∈𝑋 with𝑥∕=𝑦. Then there exist 𝐺, 𝐻in𝑇 with𝐺(𝑥)> 𝛼,𝐻(𝑦)> 𝛼and𝐺∧𝐻 ≤𝛼. Then𝛼(𝐺),𝛼(𝐻) are open sets in (𝑋, 𝑇𝛼) such that𝑥∈𝛼(𝐺), 𝑦∈𝛼(𝐻) and𝛼(𝐺)∩𝛼(𝐻) =𝛼(𝐺∩𝐻) = {𝑥∈𝑋 : (𝐺∧𝐻)(𝑥)> 𝛼}=𝜙as𝐺∧𝐻 ≤𝛼. Therefore (𝑋, 𝑇𝛼) is𝛼-Hausdorff.
Conversely, suppose (𝑋, 𝑇𝛼) is𝛼-Hausdorff. Let𝑥, 𝑦∈𝑋 with𝑥∕=𝑦. Then there exist open sets𝑈, 𝑉 in (𝑋, 𝑇𝛼) such that𝑥∈𝑈,𝑦∈𝑉 and𝑈∩𝑉 =𝜙. Let 𝑈 =𝛼(𝐺) and 𝑉 =𝛼(𝐻) for some 𝐺, 𝐻 ∈𝑇. Then 𝑥∈𝛼(𝐺) and 𝑦 ∈𝛼(𝐻).
Therefore𝐺(𝑥)> 𝛼and 𝐻(𝑦)> 𝛼. Further 𝐺∧𝐻 ≤𝛼as𝑈 ∩𝑉 =𝜙. Hence (𝑋, 𝑇) is𝛼-Hausdorff.
Let 0≤𝛼 <1 (0< 𝛼≤1). A family{𝐺𝜆:𝜆∈Λ}of fuzzy subsets of a fts (𝑋, 𝑇) is said to be an 𝛼-shading ( 𝛼∗-shading) of 𝑋 if for each𝑥∈𝑋, there exists a𝐺𝜆𝑜 in{𝐺𝜆:𝜆∈Λ}such that𝐺𝜆𝑜(𝑥)> 𝛼 (≥𝛼).
The following definition is due to Gantner et. al [3].
Definition 3.7 Let 0 ≤ 𝛼 < 1 (0 < 𝛼 ≤ 1). A fts (𝑋, 𝑇) is said to be 𝛼- compact (resp. 𝛼∗-compact) if each𝛼-shading (resp. 𝛼∗-shading) of𝑋 by open fuzzy sets has a finite𝛼-subshading (resp. 𝛼∗-subshading).
We have the following
Theorem 3.8 Let0≤𝛼 <1. A fts (𝑋, 𝑇)is𝛼-compact if and only if (𝑋, 𝑇𝛼) is compact topological space.
Proof. Let (𝑋, 𝑇) be 𝛼-compact. Let 𝑈 = {𝑈𝜆:𝜆∈Λ} be an open cover of (𝑋, 𝑇𝛼). Then, since for each𝑈𝜆, there exists a𝐺𝜆in𝑇 such that𝑈𝜆=𝛼(𝐺𝜆), we have 𝑈 ={𝛼(𝐺𝜆) :𝜆∈Λ}. Then the family 𝑉 = {𝐺𝜆:𝜆∈Λ} is an 𝛼- shading of (𝑋, 𝑇). To see this, let𝑥∈𝑋. Since𝑈 is an open cover of (𝑋, 𝑇𝛼), there is an 𝑈𝜆𝑜 ∈𝑈 such that𝑥∈𝑈𝜆𝑜. But𝑈𝜆𝑜 =𝛼(𝐺𝜆𝑜), for some𝐺𝜆𝑜 ∈𝑇. Therefore 𝑥 ∈ 𝛼(𝐺𝜆𝑜) which implies that 𝐺𝜆𝑜(𝑥) > 𝛼. By 𝛼-compactness of (𝑋, 𝑇), 𝑉 has a finite𝛼-subshading say{𝐺𝜆𝑖}𝑘𝑖=1. Then{𝛼(𝐺𝜆𝑖)}𝑘𝑖=1 forms a finite subcover of𝑈 and thus (𝑋, 𝑇𝛼) is compact.
Conversely, let (𝑋, 𝑇𝛼) be compact and 𝑈 ={𝐺𝜆:𝜆∈Λ} be an open 𝛼- shading of (𝑋, 𝑇). Then the family 𝑉 ={𝛼(𝐺𝜆) :𝜆∈Λ} is an open cover of (𝑋, 𝑇𝛼). For, let𝑥∈𝑋. Then there exists a 𝐺𝜆𝑜 in𝑈 such that𝐺𝜆𝑜(𝑥)> 𝛼.
Therefore 𝑥 ∈ 𝛼(𝐺𝜆𝑜) and (𝐺𝜆𝑜) ∈ 𝑉. By compactness of (𝑋, 𝑇𝛼), 𝑉 has a finite subcover say {𝛼(𝐺𝜆𝑖)}𝑛𝑖=1. Then the family {𝐺𝜆𝑖}𝑛𝑖=1 forms a finite 𝛼-subshading of𝑈 and hence (𝑋, 𝑇) is𝛼-compact.
Countable compact fts have been studied in [6, 11-13].
Definition 3.9 Let 0 ≤𝛼 <1 (0< 𝛼≤1). A fts (𝑋, 𝑇) is said to be count- ably𝛼-compact (resp. countably𝛼∗-compact) if every countable open𝛼-shading (resp. countable open 𝛼∗-shading) of 𝑋 has a finite 𝛼-subshading (resp. finite 𝛼∗-subshading).
It is easy to verify the following
Theorem 3.10 Let0≤𝛼 <1. A fts(𝑋, 𝑇)is countably𝛼-compact if and only if (𝑋, 𝑇𝛼)is countably compact topological space.
Lindel¨𝑜f fuzzy topological spaces were studied in [6, 8, and 12]. Lindel¨𝑜f fuzzy topological spaces, using shading families, were introduced in [16].
Definition 3.11 Let 0 ≤ 𝛼 < 1 (0 < 𝛼 ≤ 1). A fts (𝑋, 𝑇) is said to be 𝛼-Lindel¨𝑜f (resp. 𝛼∗-Lindel¨𝑜f ) if and only if every open𝛼-shading (resp. open 𝛼∗-shading) of𝑋has a countable𝛼-subshading (resp. countable𝛼∗-subshading).
Again it is easy to verify the following.
Theorem 3.12 Let0≤𝛼 <1. A fts(𝑋, 𝑇)is𝛼-Lindel¨𝑜f if and only if(𝑋, 𝑇𝛼) is Lindel¨𝑜f topological space.
Definition 3.13 Let 0 ≤𝛼 < 1 (0 < 𝛼≤ 1). Let 𝑋 be a non-empty set. A fuzzy set 𝐴in𝑋 is said to be an empty fuzzy set of order 𝛼(resp. order𝛼∗) if 𝐴(𝑥)≤𝛼(resp. 𝐴(𝑥)< 𝛼) for each𝑥∈𝑋.
A fuzzy set 𝐴 in 𝑋 is said to be non-empty of order𝛼(resp. order 𝛼∗) if there exists𝑥𝑜∈𝑋 such that 𝐴(𝑥𝑜)> 𝛼(resp. 𝐴(𝑥𝑜)≥𝛼).
Connectedness in fuzzy topological spaces was studied in [5, 9].
Connectedness, using shading families, is given in the following.
Definition 3.14 Let 0 ≤ 𝛼 < 1 (0 < 𝛼 ≤ 1). A fts (𝑋, 𝑇) is said to be 𝛼-disconnected (resp. 𝛼∗-disconnected) if there exists an𝛼-shading (resp. 𝛼∗- shading) family of two open fuzzy sets in 𝑋 which are non-empty of order 𝛼 (resp. order𝛼∗) and𝛼-disjoint (resp. 𝛼∗-disjoint).
Definition 3.15 Let 0 ≤ 𝛼 < 1 (0 < 𝛼 ≤ 1). A fts (𝑋, 𝑇) is said to be 𝛼-connected (resp. 𝛼∗-connected) if there does not exist an 𝛼-shading (resp.
𝛼∗-shading) family of two open fuzzy sets in 𝑋 which are non-empty of order𝛼 (resp. order𝛼∗) and𝛼-disjoint (resp. 𝛼∗-disjoint).
We now prove the following
Theorem 3.16 Let 0 ≤ 𝛼 < 1. A fts (𝑋, 𝑇) is 𝛼-connected if and only if (𝑋, 𝑇𝛼) is connected topological space.
Proof. Let (𝑋, 𝑇) be 𝛼-connected. Suppose (𝑋, 𝑇𝛼) is disconnected. Then there exist non-empty disjoint open sets𝑈, 𝑉 in (𝑋, 𝑇𝛼) such that𝑈 ∪𝑉 =𝑋.
Now𝑈 =𝛼(𝐺), 𝑉 =𝛼(𝐻) for some𝐺, 𝐻∈𝑇. Since𝑈, 𝑉 are non-empty sets it follows that𝐺and𝐻 are non-empty fuzzy sets of order𝛼. Further{𝐺, 𝐻} is an𝛼-shading of𝑋: For if𝑥∈𝑋 then𝑥∈𝑈 or𝑥∈𝑉 and therefore𝑥∈𝛼(𝐺) or 𝑥∈𝛼(𝐻) which implies that𝐺(𝑥)> 𝛼 or𝐻(𝑥)> 𝛼. Also𝐺, 𝐻 are𝛼-disjoint:
For,𝑈∩𝑉 =𝜙implies that𝛼(𝐺)∩𝛼(𝐻) =𝜙. Therefore𝛼(𝐺∧𝐻) =𝜙. That is {𝑥∈𝑋 : (𝐺∧𝐻)(𝑥)> 𝛼} =𝜙. Therefore for each 𝑥∈ 𝑋,(𝐺∧𝐻)(𝑥) ≤𝛼 and so𝐺, 𝐻 are𝛼-disjoint. Thus it follows that{𝐺, 𝐻}is an𝛼-shading of open fuzzy sets which are non-empty of order𝛼and are𝛼-disjoint. Therefore (𝑋, 𝑇) is𝛼-disconnected, which contradicts the hypothesis. Hence (𝑋, 𝑇𝛼) is connected topological space.
Conversely, suppose (𝑋, 𝑇𝛼) is connected. Let (𝑋, 𝑇) be 𝛼-disconnected.
Then there exist an 𝛼-shading{𝐺, 𝐻} of two open fuzzy sets in 𝑋 which are non-empty of order 𝛼 and 𝛼-disjoint. Clearly 𝛼(𝐺), 𝛼(𝐻) are open sets in (𝑋, 𝑇𝛼). Further 𝛼(𝐺), 𝛼(𝐻) are non-empty as 𝐺, 𝐻 are non-empty of order 𝛼. Also 𝛼(𝐺)∩𝛼(𝐻) = 𝛼(𝐺∧𝐻) = {𝑥∈𝑋 : (𝐺∧𝐻)(𝑥)> 𝛼} = 𝜙 since (𝐺∧𝐻)(𝑥)≤𝛼as𝐺, 𝐻 are𝛼-disjoint. Finally𝛼(𝐺)∪𝛼(𝐻) =𝑋: For if𝑥∈𝑋 then either𝐺(𝑥)> 𝛼or𝐻(𝑥)> 𝛼as {𝐺, 𝐻} is an𝛼-shading of 𝑋. Therefore 𝑥∈𝛼(𝐺) or𝑥∈𝛼(𝐻) and therefore𝑥∈𝛼(𝐺)∪𝛼(𝐻). Thus𝑋⊂𝛼(𝐺)∪𝛼(𝐻).
Also 𝛼(𝐺)∪𝛼(𝐻) ⊂ 𝑋 is obvious. Therefore 𝛼(𝐺)∪𝛼(𝐻) = 𝑋. Hence it
follows that 𝑋 is the union of two non-empty disjoint open sets in (𝑋, 𝑇𝛼) and therefore (𝑋, 𝑇𝛼) is disconnected, which contradicts the hypothesis. Hence (𝑋, 𝑇) is𝛼-connected fts.
Local compactness in fuzzy topological spaces was studied in [2, 3, 7, 14].
The definition of local compactness in [7] is modified in the following.
Definition 3.17 Let 0≤𝛼 <1 (0< 𝛼≤1). A fts (𝑋, 𝑇)is said to be locally 𝛼-compact (resp. locally 𝛼∗-compact) if for each 𝑝 ∈ 𝑋 there exists an open fuzzy set𝑁 such that 𝑁(𝑝)> 𝛼(resp. 𝑁(𝑝)≥𝛼) and 𝛼(𝑁)(resp. 𝛼∗(𝑁) ) is 𝛼-compact (resp. 𝛼∗-compact).
We prove the following
Theorem 3.18 Let 0≤𝛼 <1. A fts (𝑋, 𝑇)is locally𝛼-compact if and only if (𝑋, 𝑇𝛼)is locally compact topological space.
Proof. Let (𝑋, 𝑇) be locally 𝛼-compact. Let 𝑥 ∈ 𝑋. There exists an open fuzzy set 𝑁 in (𝑋, 𝑇) such that𝑁(𝑥)> 𝛼 and𝛼(𝑁) is 𝛼-compact. Therefore 𝛼(𝑁) is an open set in (𝑋, 𝑇𝛼) containing𝑥such that𝛼(𝑁) is compact subset in (𝑋, 𝑇𝛼): For if {𝑈𝜆=𝛼(𝐺𝜆) :𝜆∈Λ, 𝐺𝜆∈𝑇} is an open cover of𝛼(𝑁) in (𝑋, 𝑇𝛼) then the family{𝐺𝜆:𝜆∈Λ} is an open𝛼-shading of 𝛼(𝑁) in (𝑋, 𝑇).
Since𝛼(𝑁) is𝛼-compact{𝐺𝜆:𝜆∈Λ} has a finite𝛼-subshading say{𝐺𝜆𝑖}𝑘𝑖=1. Then {𝛼(𝐺𝜆𝑖) =𝑈𝜆𝑖 :𝑖= 1,2, ..., 𝑘} is a finite subcover of {𝑈𝜆:𝜆∈Λ} for 𝛼(𝑁). So 𝛼(𝑁) is a compact subset of (𝑋, 𝑇𝛼). Thus for each 𝑥 ∈ 𝑋, there exists an open set 𝛼(𝑁) in (𝑋, 𝑇𝛼) such that 𝑥∈𝛼(𝑁) and 𝛼(𝑁) is compact.
Hence (𝑋, 𝑇𝛼) is locally compact topological space.
Conversely, suppose (𝑋, 𝑇𝛼) is locally compact. Let 𝑝 ∈ 𝑋. Then there exists an open set 𝛼(𝐺) in (𝑋, 𝑇𝛼), where 𝐺 ∈ 𝑇, such that 𝑝 ∈ 𝛼(𝐺) and 𝛼(𝐺) is compact set in (𝑋, 𝑇𝛼). Now𝐺∈𝑇 and𝐺(𝑝)> 𝛼. Further𝛼(𝐺) is𝛼- compact in (𝑋, 𝑇): For if{𝐻𝜆}𝜆∈Λis an open𝛼-shading of𝛼(𝐺) in (𝑋, 𝑇), then {𝛼(𝐻𝜆) :𝜆∈Λ} is an open cover of 𝛼(𝐺). Since 𝛼(𝐺) is compact in (𝑋, 𝑇𝛼), {𝛼(𝐻𝜆) :𝜆∈Λ} has a finite subcover say {𝛼(𝐻𝜆𝑖) :𝑖= 1,2, ..., 𝑘}. Then {𝐻𝜆𝑖 :𝑖= 1,2, ..., 𝑘}is a finite𝛼-subshading of{𝐻𝜆}𝜆∈Λfor𝛼(𝐺). Therefore every open𝛼-shading for𝛼(𝐺) has a finite𝛼-subshading and therefore 𝛼(𝐺) is 𝛼-compact. Thus for each 𝑝 ∈ 𝑋 there exists an open fuzzy set 𝐺in (𝑋, 𝑇) such that𝐺(𝑝)> 𝛼 and 𝛼(𝐺) is 𝛼-compact in (𝑋, 𝑇). Hence (𝑋, 𝑇) is locally 𝛼-compact .
Acknowledgements. The authors are grateful to the referee for his com- ments which improved the readability of the paper. The authors are also grate- ful to the University Grants Commission, New Delhi for its financial support under UGC SAP I to the Department of Mathematics, Karnatak University, Dharwad India.
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S. S. Benchalli
Department of Mathematics
Karnatak University, Dharwad-580 003, Karnataka State, India.
e-mail: benchalli math@ yahoo.com
G. P. Siddapur
Department of Mathematics
Karnatak University, Dharwad-580 003, Karnataka State, India.
e-mail: siddapur math@ yahoo.com
