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Bulletin of Mathematical analysis and Applications ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 1, Issue 2, (2009), Pages 57-65 On the Level Spaces of Fuzzy Topological Spaces β S. S. Benchalli & G.P. Siddapur Abstract It is known that if (π, π ) is a fuzzy topological space and 0 β€ πΌ < 1 then the family ππΌ = {πΌ(πΊ) : πΊ β π } where πΌ(πΊ) = {π₯ β π : πΊ(π₯) > πΌ}, forms a topology on π. In the present paper some level properties have been modiο¬ed and it is proved that a fuzzy topological space (π, π ) is πΌ-compact (resp. πΌ-Hausdorο¬, countably πΌ-compact, πΌ-Lindel¨ πf, πΌconnected, locally πΌ-compact) if and only if the corresponding πΌ-level topological space (π, ππΌ ) is compact (resp. Hausdorο¬, countably compact, Lindel¨ πf, connected, locally compact). Some basic properties of πΌlevel 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 πΌ-Hausdorο¬ axiom [10], countable πΌ-compactness, πΌ-Lindel¨ πf property [6], local πΌ-compactness [7], πΌclosure [4] etc. in fuzzy topological spaces. Throughout this paper Changβs [1] deο¬nition of fuzzy topological space (abbreviated as fts) is used. If π is a set and π is a family of fuzzy subsets of π satisfying the following conditions (i) to (iii) then T is called a fuzzy topology on π ; (i) π, π β π (ii) arbitrary union of members of π is again a member of π and (iii) intersection of ο¬nitely many members of π is again a member of π . Further (π, π ) is called a fuzzy topological space (fts). If (π, π ) is a fts and 0 β€ πΌ < 1 then the family ππΌ = {πΌ(πΊ) : πΊ β π }, of all subsets of π of the form πΌ(πΊ) = {π₯ β π : πΊ(π₯) > πΌ} called πΌ-level sets, forms a topology on π [4] and is called the πΌ-level topology on π. In this paper, some basic properties of πΌ-level sets have been obtained. The πΌ-Hausdorο¬ axiom [10] and the local πΌ-compactness of [7] have been modiο¬ed. The πΌ-connectedness has been proposed. It is proved that a fts (π, π ) is πΌcompact (πΌ-Hausdorο¬, countably πΌ-compact, πΌ-Lindel¨ πf, πΌ-connected, locally πΌ-compact) if and only if the corresponding πΌ-level topological space (π, ππΌ ) is compact (resp. Hausdorο¬, countably compact, Lindel¨ πf, connected, locally compact) β Mathematics Subject Classiο¬cations 2000: 54A40, 54D05, 54D20, 54D30, 54D45.2. Key words: Fuzzy topological spaces, πΌ-level sets, πΌ-Hausdorο¬,πΌ-compact, locally πΌcompact, πΌ-Lindel¨ πf, πΌ-connected fts. c β2009 Universiteti i PrishtineΜs, PrishtineΜ, KosoveΜ. Submitted May, 2009. Published September, 2009. 57 58 2 On the Level Spaces of Fuzzy Topological Spaces πΌ-Level Sets and Their Basic Properties If πΊ is any fuzzy set in a set π and 0 β€ πΌ < 1 (0 < πΌ β€ 1) then πΌ(πΊ) = {π₯ β π : πΊ(π₯) > πΌ} (resp. πΌβ (πΊ) = {π₯ β π : πΊ(π₯) β₯ πΌ}) is called an πΌ-level (resp. πΌβ -level) set in π. The term crisp subset refers to an ordinary subset which is identiο¬ed with its characteristic function as a fuzzy subset. If π : π β π is a function and π΄ is a { fuzzy subset of π then } π (π΄) is a fuzzy subset of π deο¬ned by π (π΄)(π¦) = π π’π π΄(π₯) : π₯ β π β1 (π¦) for each π¦ β π . Further, if π΅ is a fuzzy subset of Y then π β1 (π΅) is a fuzzy subset of π deο¬ned by π β1 (π΅)(π₯) = π΅(π (π₯)) for each π₯ β π. Some basic properties of πΌ-level sets are given in the following. Theorem 2.1 Let π, π be any two sets and 0 β€ πΌ < 1. The following statements are true. 1. If πΊ is any fuzzy set in π then πΊ(π₯) β€ πΌ(πΊ)(π₯) holds for all π₯ β π with πΊ(π₯) > πΌ. 2. If πΊ β€ π» then πΌ(πΊ) β πΌ(π») for any two fuzzy sets πΊ, π» in π. 3. πΌ(πΊ) = πΊ if and only if πΊ is a crisp susbet of π. 4. πΌ(πΌ(πΊ)) = πΌ(πΊ) for any fuzzy set πΊ in π. β βͺ 5. πΌ( πΊπ ) = πΌ(πΊπ ) for any family {πΊπ : π β Ξ} of fuzzy sets in π. π π β β© 6. πΌ( πΊπ ) = πΌ(πΊπ ) for any family {πΊπ : π β Ξ} of fuzzy sets in π. π π 7. If π : π β π , then π (πΌ(πΊ)) = πΌ(π (πΊ)) for each fuzzy set πΊ in π. 8. If π : π β π , then π β1 (πΌ(πΊ)) = πΌ(π β1 (πΊ)) for each fuzzy set πΊ in π . 9. πΌ(πΊ × π») = πΌ(πΊ) × πΌ(π») for any two fuzzy sets πΊ, π» in π where πΊ × π» is a fuzzy set in π × π given by (πΊ × π»)(π₯, π¦) = πΊ(π₯) β§ π»(π¦) for each (π₯, π¦) β π × π . Proof. (1). Let π₯ β π with πΊ(π₯) > πΌ. Then π₯ β πΌ(πΊ) so that (πΌ(πΊ))(π₯) = 1 β₯ πΊ(π₯) > πΌ and therefore πΊ(π₯) β€ (πΌ(πΊ))(π₯). (2) If π₯ β πΌ(πΊ) then πΊ(π₯) > π and therefore π»(π₯) β₯ πΊ(π₯) > πΌ. Consequently π₯ β πΌ(π»). (3) If πΊ is crisp and if π₯ β π then πΊ(π₯) = 0 or 1. If πΊ(π₯) = 0 then π₯ β / πΌ(πΊ) and therefore (πΌ(πΊ))(π₯) = 0 which proves πΊ(π₯) = πΌ(πΊ(π₯)). In case if πΊ(π₯) = 1, then πΊ(π₯) = 1 > πΌ and therefore π₯ β πΌ(πΊ) which proves (πΌ(πΊ))(π₯) = 1 = πΊ(π₯). The converse part follows as πΌ(πΊ) is crisp. S.S.Benchalli & G. P. Siddapur 59 (4) Follows from (3) as πΌ(πΊ) is crisp. β (5) If π₯ β πΌ( πΊπ ) then ππ’π(πΊπ (π₯)) > πΌ. Consequently there exists a ππ π such that πΊππ (π₯) > πΌ which implies π₯ β πΌ(πΊππ ) and hence π₯ β βͺ πΌ(πΊπ ). π β βͺ βͺ β Therefore πΌ( πΊπ ) β πΌ(πΊπ ). Similarly πΌ(πΊπ ) β πΌ( πΊπ ) and hence the π π π π equality. (6) If π₯ β πΌ( β πΊπ ) then ( π β πΊπ )(π₯) > πΌ and therefore πΊπ (π₯) > πΌ for each π π. This implies that π₯ β πΌ(πΊπ ) for each π and therefore π₯ β β πΌ(πΊπ ). Thus π β β© β© β πΌ( πΊπ ) β πΌ(πΊπ ). Similarly πΌ(πΊπ ) β πΌ( πΊπ ). π π π π (7) If π¦ β π (πΌ(πΊ)) then there is an{element π₯ β πΌ(πΊ)} such that π¦ = π (π₯). Now πΊ(π₯) > πΌ and therefore ππ’π πΊ(π₯) : π₯ β π β1 (π¦) > πΌ which implies (π (πΊ))(π¦) > πΌ. Then π¦ β πΌ(π (πΊ)). Thus π (πΌ(πΊ)) β πΌ(π (πΊ)). Similarly it can be shown that πΌ(π (πΊ)) β π (πΌ(πΊ)) and hence the result follows. β1 (8) Let π₯ β Then π (π₯) = π¦ β πΌ(πΊ) [so that πΊ(π¦) = πΊ(π (π₯)) > πΌ. [ πβ1 (πΌ(πΊ)). ] ] Therefore π (πΊ) (π₯) > πΌ which implies π₯ β πΌ π β1 (πΊ) and hence it follows that π β1 (πΌ(πΊ)) β πΌ(π β1 (πΊ)). Similarly πΌ(π β1 (πΊ)) β π β1 (πΌ(πΊ)) and hence the equality. (9) If (π₯, π¦) β πΌ(πΊ × π») then (πΊ × π»)(π₯, π¦) > πΌ and therefore π₯ β πΌ(πΊ) and π¦ β πΌ(π»). So (π₯, π¦) β πΌ(πΊ) × πΌ(π»). Thus πΌ(πΊ × π») β πΌ(πΊ) × πΌ(π»). Similarly it can be shown that πΌ(πΊ) × πΌ(π») β πΌ(πΊ × π») and hence the equality follows. 3 Level Spaces and Main Results In the beginning of this section we deal with Rodabaughβs [10] πΌ-Hausdorο¬ fts. Deο¬nition 3.1 Let 0 β€ πΌ < 1 (0 < πΌ β€ 1). A fts (π, π ) is said to be πΌHausdorο¬ (resp. πΌβ -Hausdorο¬ ) if for each π₯, π¦ in π with π₯ β= π¦, there exist πΊ, π» in π such that πΊ(π₯) > πΌ (resp. πΊ(π₯) β₯ πΌ), π»(π¦) > πΌ (resp. π»(π¦) β₯ πΌ) and πΊ β§ π» = 0. We have the following Theorem 3.2 Let 0 β€ πΌ < 1. If a fts (π, π ) is πΌ-Hausdorο¬, then (π, ππΌ ) is Hausdorο¬ topological space. 60 On the Level Spaces of Fuzzy Topological Spaces Proof. Let π₯, π¦ β π with π₯ β= π¦. Then there are πΊ, π» in π such that πΊ(π₯) > πΌ, π»(π¦) > πΌ and πΊ β§ π» = 0. Then πΌ(πΊ) and πΌ(π») are open sets in (π, ππΌ ) and π₯ β πΌ(πΊ), π¦ β πΌ(π»). Also πΌ(πΊ) β© πΌ(π») = π since πΊ β§ π» = 0. Hence (π, ππΌ ) is Hausdorο¬ topological space. The converse of the above theorem holds for the case of πΌ = 0, which is given in the following. Theorem 3.3 Let (π, π ) be a fts. If (π, π0 ) is Hausdorο¬ topological space, then (π, π ) is 0-Hausdorο¬ fts. Proof. Let π₯, π¦ β π with π₯ β= π¦. Then there are open sets π, π in (π, π0 ) such that π₯ β π , π¦ β π and π β© π = π. Let π = 0(πΊ), π = 0(π») for some πΊ, π» in π . Then it follows that πΊ(π₯) > 0 and π»(π¦) > 0. Further πΊ β§ π» = 0 as π β© π = π. Hence (π, π ) is 0-Hausdorο¬. Deο¬nition 3.4 Let π be a set and 0 β€ πΌ < 1 (0 < πΌ β€ 1). A family {πΊπ }π of β πΊπ β€ πΌ (resp. fuzzy sets in π is said to be πΌ-disjoint (resp. πΌβ -disjoint) if π β πΊπ < πΌ). π It is evident that two fuzzy sets πΊ, π» in π are πΌ-disjoint (πΌβ -disjoint) if and only if for each π₯ in π either πΊ(π₯) β€ πΌ (resp. πΊ(π₯) < πΌ) or π»(π₯) β€ πΌ (resp.π»(π₯) < πΌ). Rodabaughβs deο¬nition is suitably modiο¬ed in the following. Deο¬nition 3.5 Let 0 β€ πΌ < 1 (0 < πΌ β€ 1). A fts (π, π ) is said to be πΌHausdorο¬ (resp. πΌβ -Hausdorο¬ ) if for each π₯, π¦ in π with π₯ β= π¦, there exist πΊ, π» in π such that πΊ(π₯) > πΌ (resp. πΊ(π₯) β₯ πΌ), π»(π¦) > πΌ (resp. π»(π¦) β₯ πΌ) and πΊ, π» are πΌ-disjoint (resp. πΌβ -disjoint). For the modiο¬ed class of πΌ-Hausdorο¬ fuzzy topological spaces we have the following. Theorem 3.6 Let 0 β€ πΌ < 1. A fts (π, π ) is a πΌ-Hausdorο¬ if and only if (π, ππΌ ) is Hausdorο¬ topological space. Proof. Let (π, π ) be πΌ-Hausdorο¬. Let π₯, π¦ β π with π₯ β= π¦. Then there exist πΊ, π» in π with πΊ(π₯) > πΌ, π»(π¦) > πΌ and πΊβ§π» β€ πΌ. Then πΌ(πΊ), πΌ(π») are open sets in (π, ππΌ ) such that π₯ β πΌ(πΊ), π¦ β πΌ(π») and πΌ(πΊ) β© πΌ(π») = πΌ(πΊ β© π») = {π₯ β π : (πΊ β§ π»)(π₯) > πΌ} = π as πΊ β§ π» β€ πΌ. Therefore (π, ππΌ ) is πΌ-Hausdorο¬. Conversely, suppose (π, ππΌ ) is πΌ-Hausdorο¬. Let π₯, π¦ β π with π₯ β= π¦. Then there exist open sets π, π in (π, ππΌ ) such that π₯ β π , π¦ β π and π β©π = π. Let π = πΌ(πΊ) and π = πΌ(π») for some πΊ, π» β π . Then π₯ β πΌ(πΊ) and π¦ β πΌ(π»). Therefore πΊ(π₯) > πΌ and π»(π¦) > πΌ. Further πΊ β§ π» β€ πΌ as π β© π = π. Hence (π, π ) is πΌ-Hausdorο¬. S.S.Benchalli & G. P. Siddapur 61 Let 0 β€ πΌ < 1 (0 < πΌ β€ 1). A family {πΊπ : π β Ξ} of fuzzy subsets of a fts (π, π ) is said to be an πΌ-shading ( πΌβ -shading) of π if for each π₯ β π, there exists a πΊππ in {πΊπ : π β Ξ} such that πΊππ (π₯) > πΌ (β₯ πΌ). The following deο¬nition is due to Gantner et. al [3]. Deο¬nition 3.7 Let 0 β€ πΌ < 1 (0 < πΌ β€ 1). A fts (π, π ) is said to be πΌcompact (resp. πΌβ -compact) if each πΌ-shading (resp. πΌβ -shading) of π by open fuzzy sets has a ο¬nite πΌ-subshading (resp. πΌβ -subshading). We have the following Theorem 3.8 Let 0 β€ πΌ < 1. A fts (π, π ) is πΌ-compact if and only if (π, ππΌ ) is compact topological space. Proof. Let (π, π ) be πΌ-compact. Let π = {ππ : π β Ξ} be an open cover of (π, ππΌ ). Then, since for each ππ , there exists a πΊπ in π such that ππ = πΌ(πΊπ ), we have π ={πΌ(πΊπ ) : π β Ξ}. Then the family π = {πΊπ : π β Ξ} is an πΌshading of (π, π ). To see this, let π₯ β π. Since π is an open cover of (π, ππΌ ), there is an πππ β π such that π₯ β πππ . But πππ = πΌ(πΊππ ), for some πΊππ β π . Therefore π₯ β πΌ(πΊππ ) which implies that πΊππ (π₯) > πΌ. By πΌ-compactness of π π (π, π ), π has a ο¬nite πΌ-subshading say {πΊππ }π=1 . Then {πΌ(πΊππ )}π=1 forms a ο¬nite subcover of π and thus (π, ππΌ ) is compact. Conversely, let (π, ππΌ ) be compact and π ={πΊπ : π β Ξ} be an open πΌshading of (π, π ). Then the family π = {πΌ(πΊπ ) : π β Ξ} is an open cover of (π, ππΌ ). For, let π₯ β π. Then there exists a πΊππ in π such that πΊππ (π₯) > πΌ. Therefore π₯ β πΌ(πΊππ ) and (πΊππ ) β π . By compactness of (π, ππΌ ), π has π π a ο¬nite subcover say {πΌ(πΊππ )}π=1 . Then the family {πΊππ }π=1 forms a ο¬nite πΌ-subshading of π and hence (π, π ) is πΌ-compact. Countable compact fts have been studied in [6, 11-13]. Deο¬nition 3.9 Let 0 β€ πΌ < 1 (0 < πΌ β€ 1). A fts (π, π ) is said to be countably πΌ-compact (resp. countably πΌβ -compact) if every countable open πΌ-shading (resp. countable open πΌβ -shading) of π has a ο¬nite πΌ-subshading (resp. ο¬nite πΌβ -subshading). It is easy to verify the following Theorem 3.10 Let 0 β€ πΌ < 1. A fts (π, π ) is countably πΌ-compact if and only if (π, ππΌ ) is countably compact topological space. Lindel¨ πf fuzzy topological spaces were studied in [6, 8, and 12]. Lindel¨ πf fuzzy topological spaces, using shading families, were introduced in [16]. Deο¬nition 3.11 Let 0 β€ πΌ < 1 (0 < πΌ β€ 1). A fts (π, π ) is said to be πΌ-Lindel¨ πf (resp. πΌβ -Lindel¨ πf ) if and only if every open πΌ-shading (resp. open β πΌ -shading) of π has a countable πΌ-subshading (resp. countable πΌβ -subshading). Again it is easy to verify the following. 62 On the Level Spaces of Fuzzy Topological Spaces Theorem 3.12 Let 0 β€ πΌ < 1. A fts (π, π ) is πΌ-Lindel¨ πf if and only if (π, ππΌ ) is Lindel¨ πf topological space. Deο¬nition 3.13 Let 0 β€ πΌ < 1 (0 < πΌ β€ 1). Let π be a non-empty set. A fuzzy set π΄ in π is said to be an empty fuzzy set of order πΌ (resp. order πΌβ ) if π΄(π₯) β€ πΌ (resp. π΄(π₯) < πΌ) for each π₯ β π. A fuzzy set π΄ in π is said to be non-empty of order πΌ (resp. order πΌβ ) if there exists π₯π β π such that π΄(π₯π ) > πΌ (resp. π΄(π₯π ) β₯ πΌ). Connectedness in fuzzy topological spaces was studied in [5, 9]. Connectedness, using shading families, is given in the following. Deο¬nition 3.14 Let 0 β€ πΌ < 1 (0 < πΌ β€ 1). A fts (π, π ) is said to be πΌ-disconnected (resp. πΌβ -disconnected) if there exists an πΌ-shading (resp. πΌβ shading) family of two open fuzzy sets in π which are non-empty of order πΌ (resp. order πΌβ ) and πΌ-disjoint (resp. πΌβ -disjoint). Deο¬nition 3.15 Let 0 β€ πΌ < 1 (0 < πΌ β€ 1). A fts (π, π ) is said to be πΌ-connected (resp. πΌβ -connected) if there does not exist an πΌ-shading (resp. πΌβ -shading) family of two open fuzzy sets in π which are non-empty of order πΌ (resp. order πΌβ ) and πΌ-disjoint (resp. πΌβ -disjoint). We now prove the following Theorem 3.16 Let 0 β€ πΌ < 1. A fts (π, π ) is πΌ-connected if and only if (π, ππΌ ) is connected topological space. Proof. Let (π, π ) be πΌ-connected. Suppose (π, ππΌ ) is disconnected. Then there exist non-empty disjoint open sets π, π in (π, ππΌ ) such that π βͺ π = π. Now π = πΌ(πΊ), π = πΌ(π») for some πΊ, π» β π . Since π, π are non-empty sets it follows that πΊ and π» are non-empty fuzzy sets of order πΌ. Further {πΊ, π»} is an πΌ-shading of π: For if π₯ β π then π₯ β π or π₯ β π and therefore π₯ β πΌ(πΊ) or π₯ β πΌ(π») which implies that πΊ(π₯) > πΌ or π»(π₯) > πΌ. Also πΊ, π» are πΌ-disjoint: For, π β© π = π implies that πΌ(πΊ) β© πΌ(π») = π. Therefore πΌ(πΊ β§ π») = π. That is {π₯ β π : (πΊ β§ π»)(π₯) > πΌ} = π. Therefore for each π₯ β π,(πΊ β§ π»)(π₯) β€ πΌ and so πΊ, π» are πΌ-disjoint. Thus it follows that {πΊ, π»} is an πΌ-shading of open fuzzy sets which are non-empty of order πΌ and are πΌ-disjoint. Therefore (π, π ) is πΌ-disconnected, which contradicts the hypothesis. Hence (π, ππΌ ) is connected topological space. Conversely, suppose (π, ππΌ ) is connected. Let (π, π ) be πΌ-disconnected. Then there exist an πΌ-shading {πΊ, π»} of two open fuzzy sets in π which are non-empty of order πΌ and πΌ-disjoint. Clearly πΌ(πΊ), πΌ(π») are open sets in (π, ππΌ ). Further πΌ(πΊ), πΌ(π») are non-empty as πΊ, π» are non-empty of order πΌ. Also πΌ(πΊ) β© πΌ(π») = πΌ(πΊ β§ π») = {π₯ β π : (πΊ β§ π»)(π₯) > πΌ} = π since (πΊ β§ π»)(π₯) β€ πΌ as πΊ, π» are πΌ-disjoint. Finally πΌ(πΊ) βͺ πΌ(π») = π: For if π₯ β π then either πΊ(π₯) > πΌ or π»(π₯) > πΌ as {πΊ, π»} is an πΌ-shading of π. Therefore π₯ β πΌ(πΊ) or π₯ β πΌ(π») and therefore π₯ β πΌ(πΊ) βͺ πΌ(π»). Thus π β πΌ(πΊ) βͺ πΌ(π»). Also πΌ(πΊ) βͺ πΌ(π») β π is obvious. Therefore πΌ(πΊ) βͺ πΌ(π») = π. Hence it S.S.Benchalli & G. P. Siddapur 63 follows that π is the union of two non-empty disjoint open sets in (π, ππΌ ) and therefore (π, ππΌ ) is disconnected, which contradicts the hypothesis. Hence (π, π ) is πΌ-connected fts. Local compactness in fuzzy topological spaces was studied in [2, 3, 7, 14]. The deο¬nition of local compactness in [7] is modiο¬ed in the following. Deο¬nition 3.17 Let 0 β€ πΌ < 1 (0 < πΌ β€ 1). A fts (π, π ) is said to be locally πΌ-compact (resp. locally πΌβ -compact) if for each π β π there exists an open fuzzy set π such that π (π) > πΌ (resp. π (π) β₯ πΌ) and πΌ(π ) (resp. πΌβ (π ) ) is πΌ-compact (resp. πΌβ -compact). We prove the following Theorem 3.18 Let 0 β€ πΌ < 1. A fts (π, π ) is locally πΌ-compact if and only if (π, ππΌ ) is locally compact topological space. Proof. Let (π, π ) be locally πΌ-compact. Let π₯ β π. There exists an open fuzzy set π in (π, π ) such that π (π₯) > πΌ and πΌ(π ) is πΌ-compact. Therefore πΌ(π ) is an open set in (π, ππΌ ) containing π₯ such that πΌ(π ) is compact subset in (π, ππΌ ): For if {ππ = πΌ(πΊπ ) : π β Ξ, πΊπ β π } is an open cover of πΌ(π ) in (π, ππΌ ) then the family {πΊπ : π β Ξ} is an open πΌ-shading of πΌ(π ) in (π, π ). π Since πΌ(π ) is πΌ-compact {πΊπ : π β Ξ} has a ο¬nite πΌ-subshading say {πΊππ }π=1 . Then {πΌ(πΊππ ) = πππ : π = 1, 2, ......, π} is a ο¬nite subcover of {ππ : π β Ξ} for πΌ(π ). So πΌ(π ) is a compact subset of (π, ππΌ ). Thus for each π₯ β π, there exists an open set πΌ(π ) in (π, ππΌ ) such that π₯ β πΌ(π ) and πΌ(π ) is compact. Hence (π, ππΌ ) is locally compact topological space. Conversely, suppose (π, ππΌ ) is locally compact. Let π β π. Then there exists an open set πΌ(πΊ) in (π, ππΌ ), where πΊ β π , such that π β πΌ(πΊ) and πΌ(πΊ) is compact set in (π, ππΌ ). Now πΊ β π and πΊ(π) > πΌ. Further πΌ(πΊ) is πΌcompact in (π, π ): For if {π»π }πβΞ is an open πΌ-shading of πΌ(πΊ) in (π, π ), then {πΌ(π»π ) : π β Ξ} is an open cover of πΌ(πΊ). Since πΌ(πΊ) is compact in (π, ππΌ ), {πΌ(π»π ) : π β Ξ} has a ο¬nite subcover say {πΌ(π»ππ ) : π = 1, 2, ......., π}. Then {π»ππ : π = 1, 2, ......., π} is a ο¬nite πΌ-subshading of {π»π }πβΞ for πΌ(πΊ). Therefore every open πΌ-shading for πΌ(πΊ) has a ο¬nite πΌ-subshading and therefore πΌ(πΊ) is πΌ-compact. Thus for each π β π there exists an open fuzzy set πΊ in (π, π ) such that πΊ(π) > πΌ and πΌ(πΊ) is πΌ-compact in (π, π ). Hence (π, π ) is locally πΌ-compact . Acknowledgements. The authors are grateful to the referee for his comments which improved the readability of the paper. 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Benchalli Department of Mathematics Karnatak University, Dharwad-580 003, Karnataka State, India. e-mail: benchalli math@ yahoo.com S.S.Benchalli & G. P. Siddapur G. P. Siddapur Department of Mathematics Karnatak University, Dharwad-580 003, Karnataka State, India. e-mail: siddapur math@ yahoo.com 65