sets and continuity in fuzzy topological spaces

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1 Journal of Linar and Topological Algbra Vol. 06, No. 02, 2017, Fuzzy sts and continuity in fuzzy topological spacs A. Vadivl a, B. Vijayalakshmi b a Dpartmnt of Mathmatics, Annamalai Univrsity, Annamalai Nagar, Tamil Nadu , India. b Mathmatics Sction, FEAT, Annamalai Univrsity, Annamalai Nagar, Tamil Nadu , India. Rcivd 5 April 2017; Rvisd 1 July 2017; Accptd 3 Sptmbr Abstract. W introduc a nw class of fuzzy opn sts calld fuzzy sts which includs th class of fuzzy -opn sts. W also dfin a wakr form of fuzzy sts trmd as fuzzy locally sts. By mans of ths nw sts, w prsnt th notions of fuzzy continuity and fuzzy locally continuity which ar wakr than fuzzy -continuity and furthrmor w invstigat th rlationships btwn ths nw typs of continuity and som othrs. c 2017 IAUCTB. All rights rsrvd. Kywords: Fuzzy st, fuzzy st, fuzzy locally st, fuzzy continuity, fuzzy locally continuity AMS Subjct Classification: Primary 54A40, 54C05; Scondary 03E Introduction Th concpts of fuzzy sts and fuzzy topology wr firstly givn by Zadh in [26] and Chang in [5], and aftr thn thr hav bn many dvlopmnts on dfining uncrtain situations and rlations in mor ralistic way. Th fuzzy topology thory has rapidly bgan to play an important rol in many diffrnt scintific aras such as conomics, quantum physics and gographic information systm (GIS). For instanc, Shi and Liu mntiond that th fuzzy topology thory can potntially provid a mor ralistic dscription of uncrtain spatial objcts and uncrtain rlations in [25] whr thy dvlopd th computational fuzzy topology Which is basd on th intrior and th closur oprator. Corrsponding author. addrss: mathvijaya2006au@gmail.com (B. Vijayalakshmi). Print ISSN: c 2017 IAUCTB. All rights rsrvd. Onlin ISSN:

2 126 A. Vadivl t al. / J. Linar. Topological. Algbra. 06(02) (2017) Bsids, th concpts of fuzzy topology and fuzzy sts hav vry important applications on particl physic in connction with string thory and ϵ thory studid by El-Naschi [18], [19], [20]. Maki [15] introducd th notion of sts in topological spacs. A st is a st λ which is qual to to its krnl (saturatd sts), (i.) to th intrsction of all opn suprsts of λ. Arnas t al.[2] introducd and invstigatd th notion of λ-closd sts and λ-opn sts by involving -sts and closd sts. In 2008, Erdal Ekici [7 11], has introducd and studid th concpt of -opn sts in gnral topology. Snivasan [23] dfind th concpt of fuzzy -opn sts and studid fuzzy -continuous mappings on fuzzy topological spacs. Fuzzy opn sts ar wakr than fuzzy δ propn st, fuzzy δ smi opn sts. Using ths notion, h studid fuzzy -continuous mappings in fuzzy topological spacs. In this papr, w xtnd th notion of -opn sts to fuzzy topological spac in th nam fuzzy sts and fuzzy locally sts and study som proprtis basd on this concpt. W also introduc th concpts of fuzzy continuity and fuzzy locally continuity. 2. Prliminaris Throughout this papr, nonmpty sts will b dnotd by X, Y tc., I = [0, 1] and I 0 = (0, 1]. For α I, α(x) = { α for all x X. A fuzzy point x t [14] for t I 0 is an lmnt of I X t if y = x such that x t (y) = Th st of all fuzzy points in X is dnotd 0 if y / x. by P t(x). A fuzzy point x t λ [14] iff t < λ(x). A fuzzy st λ is quasi-coincidnt with µ [14], dnotd by λqµ, if thr xists x X such that λ(x) + µ(x) > 1. If λ is not quasi-coincidnt with { µ, w dnotd λ qµ. If A X, w dfin th charactristic function 1 if x A, χ A on X by χ A (x) = All othr notations and dfinitions ar standard, for 0 if x / A. all in th fuzzy st thory. Hr, (X, τ) man fuzzy topological spac (fts, for short) in Chang s sns [5]. For a fuzzy st λ of a fts X, th notion I X, λ c = 1 X λ, Cl(λ), Int(λ), will rspctivly stand for th st of all fuzzy substs of X, th complmnt, fuzzy closur, fuzzy intrior, of λ. By 1 ϕ (or 0 X or ϕ) and 1 X (or X) w will man th fuzzy null st and fuzzy whol st with constant mmbrship function 0 (zro function) and 1 (unit function) rspctivly. Dfinition 2.1 A fuzzy subst λ in a fuzzy topological spac (X, τ) is calld (i) fuzzy rgular opn (rsp. fuzzy rgular closd) st [1] if Int(Cl(λ)) = λ (rsp. Cl(Int(λ)) = λ) or if 1 X λ is fuzzy rgular opn st in X. (ii) fuzzy δ propn st [3] if λ Int(δCl(λ)) (rsp. fuzzy δ prclosd st) if λ Cl(δInt(λ)). (iii) fuzzy δ smiopn st [17] if λ Cl(δInt(λ)) (rsp. fuzzy δ smiclosd st) if λ Int(δCl(λ)). (iv) a fuzzy -opn st [23] of X if λ Cl(δInt(λ)) Int(δCl(λ)). (v) a fuzzy -closd st [23] of X if Cl(δInt(λ)) Int(δCl(λ)) λ. Th family of all -opn (rsp. -closd) sts of X will b dnotd by O(X) (rsp. C(X).) Dfinition 2.2 [23] Lt (X, τ) b a fuzzy topological spac. Lt λ b a fuzzy st of a fuzzy topological spac X.

3 A. Vadivl t al. / J. Linar. Topological. Algbra. 06(02) (2017) (i) Int(λ) = {µ I X : µ λ, µ is a O st } is calld th fuzzy -intrior of λ. (ii) Cl(λ) = {µ I X : µ λ, µ is a C st } is calld th fuzzy -closur of λ. Dfinition 2.3 Lt f : (X, τ 1 ) (Y, τ 2 ) b a mapping from a fts (X, τ 1 ) to anothr (Y, τ 2 ). Thn f is calld (1) fuzzy continuous [5] if f 1 (λ) is fuzzy opn st in X for any fuzzy opn st λ in Y. (2) fuzzy δ-smicontinuous [6] if f 1 (µ) is a fuzzy smiopn st of X for ach µ τ 2. (3) fuzzy δ-prcontinuous [3] if f 1 (µ) is a fuzzy pr-opn st of X for ach µ τ 2. (4) fuzzy -continuous [23] if f 1 (µ) is a fuzzy -opn st of X for ach µ τ Fuzzy and fuzzy sts In this sction, w dfin th class of fuzzy sts which is a wakr form of fuzzy -opn sts and invstigat som basic proprtis of this class. W also prsnt th fuzzy sts as th dual concpt of fuzzy sts. Dfinition 3.1 Lt (X, τ) b a fuzzy topological spac and λ b a fuzzy st of X. Th fuzzy st λ and λ st of λ ar dfind as follows: λ } = {α : λ α, α O(X), λ } = {γ : γ λ, γ C(X). Proposition 3.2 Lt (X, τ) b a fuzzy topological spac and λ, µ and µ i (i Ω) b th fuzzy sts of X. Th following statmnts ar valid: (i) µ µ, (ii) If λ µ, thn λ µ, (iii) (µ ) = µ, (iv) µ i ( µ i ), (v) ( µ i ) µ i, (vi) µ µ, (vii) If λ µ, thn λ µ, (viii) (µ ) = µ, (ix) µ i ( µ i ), (x) ( µ i ) µ i, (xi) (µ c ) = (µ ) c. Proof. W will prov only (v) and (xi). Th othrs can b provd in a similar way. For all i Ω w hav µ i µ i ( ) µ i ( µ i ) ( ) µ i µ i.

4 128 A. Vadivl t al. / J. Linar. Topological. Algbra. 06(02) (2017) which provs (v). For (xi), (µ ( { }) c ) c = γ : γ µ, γ C(X) = } {γ c : µ c γ c, γ c O(X) = } {α : µ c α, α O(X) = (µ c ) Dfinition 3.3 Lt λ b a fuzzy st of a fuzzy topological spac (X, τ). Thn λ is calld (1) a fuzzy st st if λ = λ. (2) a fuzzy st st if λ = λ. Th family of all fuzzy sts and sts will b dnotd by (X) and (X), rspctivly. Thorm 3.4 µ is a fuzzy st iff µc is a fuzzy st. Proof. It is obvious. Proposition 3.5 Lt λ b fuzzy st of a fuzzy topological spac (X, τ). (i) If λ O(X), thn λ (X). (ii) If λ C(X), thn λ (X). Proof. It is obvious. Rmark 1 Non of th rvrs implications in Proposition 3.5 is valid as shown in th following xampls. Exampl 3.6 Lt λ and µ b fuzzy substs of X = {a, b} and dfind as follows: λ(a) = 0.2, λ(b) = 0.3; µ(a) = 0.1, µ(b) = Thn, τ = {0, 1, λ} is a fuzzy topology on X. Clarly, it can b shown that Cl(δIntµ) = 0 and Int(δClµ) = λ. Sinc, µ Cl(δIntµ) Int(δClµ) = (0.2 a, 0.3 b ), µ is not a fuzzy -opn st. Howvr, µ = µ. Thrfor, µ is a fuzzy st. Exampl 3.7 Lt λ and µ b fuzzy substs of X = {a, b} and dfind as follows: λ(a) = 0.3, λ(b) = 0.5; µ(a) = 0.6, µ(b) = Thn, τ = {0, 1, λ} is a fuzzy topology on X. Clarly, it can b shown that Cl(δIntµ) = λ c and Int(δClµ) = 1. Sinc, µ Cl(δIntµ) Int(δClµ) = (0.7 a, 0.5 b ), µ is not a fuzzy -closd st. Howvr, µ = µ. Thrfor, µ is a fuzzy st. Exampl 3.8 Lt λ, µ and ω b fuzzy substs of X = {a, b, c} and dfind as follows: λ(a) = 0.2, λ(b) = 0.3, λ(c) = 0.4; µ(a) = 0.1, µ(b) = 0.1, µ(c) = 0.4; ω(a) = 0.2, ω(b) = 0.4, ω(c) = 0.4. Thn, τ = {0, 1, λ, µ} is a fuzzy topology on X. Sinc, ω = (0.2 a, 0.4 b, 0.4 c ) Int(δCl(ω)) = λ = (0.2 a, 0.3 a, 0.4 a ), ω is not fuzzy δ propn. On th othr hand, ω = ω. That is ω is a fuzzy st.

5 A. Vadivl t al. / J. Linar. Topological. Algbra. 06(02) (2017) Exampl 3.9 Lt λ, µ and ω b fuzzy substs of X = {a, b, c} and dfind as follows: λ(a) = 0.3, λ(b) = 0.4, λ(c) = 0.5; µ(a) = 0.6, µ(b) = 0.5, µ(c) = 0.5; ω(a) = 0.7, ω(b) = 0.7, ω(c) = 0.5. Thn, τ = {0, 1, λ, µ} is a fuzzy topology on X. Sinc, ω = (0.7 a, 0.7 b, 0.5 c ) Int(δCl(ω)) = λ c = (0.7 a, 0.6 a, 0.5 a ), ω is not fuzzy δ smiopn. On th othr hand, ω = ω. That is ω is a fuzzy st. Exampl 3.10 Lt λ, µ and ω b fuzzy substs of X = {a, b, c} and dfind as follows: λ(a) = 0.3, λ(b) = 0.5, λ(c) = 0.2; µ(a) = 0.4, µ(b) = 0.5, µ(c) = 0.5; ω(a) = 0.2, ω(b) = 0.2, ω(c) = 0.2. Thn, τ = {0, 1, λ, µ} is a fuzzy topology on X. Sinc, ω Int(δCl(ω)) and ω Cl(δInt(ω)), ω is fuzzy δ propn and fuzzy δ smiopn. But ω is not fuzzy opn. Rmark 2 Evry fuzzy st is fuzzy locally shown in Exampl 4.2. st but th convrs is not tru as Rmark 3 Th following diagram of th implications is tru. fuzzy opn fuzzy δ-propn fuzzy δ-smiopn fuzzy -opn fuzzy st fuzzy locally st Thorm 3.11 Lt λ and λ i (i Ω) b th fuzzy sts of th fuzzy topological spac (X, τ). Thn (1) λ is a fuzzy st. (2) λ is a fuzzy st. (3) If {λ i : i Ω} (X), thn λ i is a fuzzy st. (4) If {λ i : i Ω} (X), thn λ i is a fuzzy st. Proof. (1) By Proposition 3.2(iii), (λ ) (2) It is clar from Proposition 3.2 (viii). (3) For all i Ω, w hav Sinc for all i Ω, = λ. Hnc λ is a fuzzy st. (λ i ) = λi λi = λ i ( λ i) (λ i ) = λ i. λ i ( λ i ) holds, thus λ i = ( λ i ). (4) It can b provd in similar mannr in (3).

6 130 A. Vadivl t al. / J. Linar. Topological. Algbra. 06(02) (2017) Fuzzy locally sts In this sction, w introduc th class of fuzzy locally sts including th class of fuzzy sts and giv two charactrizations of ths sts. Dfinition 4.1 Lt λ b a fuzzy st of a fuzzy topological spac (X, τ). λ is calld fuzzy locally st if thr xists a fuzzy st α and a fuzzy -closd st β such that λ = α β. Rmark 4 Sinc, λ = λ 1 X, for vry fuzzy st λ, vry fuzzy st and vry fuzzy -closd st is a fuzzy locally st. st is a fuzzy locally Exampl 4.2 Lt λ, µ, γ and δ b fuzzy substs of X = {a, b} and dfind as follows: λ(a) = 0.2, λ(b) = 0; µ(a) = 0.1, µ(b) = 1; γ(a) = 0.2, γ(b) = 1. Thn, τ = {0, 1, λ} is a fuzzy topology on X. Clarly, it can b shown that Cl(δIntµ) = 0 and Int(δClµ) = λ. Sinc, µ Cl(δIntµ) Int(δClµ) = λ = (0.2 a, 0.0 b ), µ is not a fuzzy -opn st. Hnc, µ / µ (X). But µ is a fuzzy locally st, sinc µ can b rprsntd as µ = γ µ whr γ is a fuzzy st and µ is a fuzzy -closd st. Hnc, fuzzy locally st nd not b fuzzy st. Rmark 5 Evry fuzzy -closd st is fuzzy locally tru as shown in xampl blow. st but th convrs nd not b Exampl 4.3 Lt λ, µ, γ and δ b fuzzy substs of X = {a, b} and dfind as follows: λ(a) = 0.3, λ(b) = 0.5; µ(a) = 0.6, µ(b) = 0.7; δ(a) = 0.7, δ(b) = 0.9; Thn, τ = {0, 1, λ} is a fuzzy topology on X. Clarly, it can b shown that Cl(δIntµ) = λ c and Int(δClµ) = 1. Sinc, µ Cl(δIntµ) Int(δClµ) = λ c = (0.7 a, 0.5 b ), µ is not a fuzzy -closd st. But µ is a fuzzy locally st, sinc µ can b rprsntd as µ = µδ whr µ is a fuzzy st and δ is a fuzzy -closd st. Hnc, fuzzy locally st nd not b fuzzy -closd st. Thorm 4.4 Lt λ b a fuzzy st of a fuzzy topological spac (X, τ). Th following statmnts ar quivalnt: (1) λ is a fuzzy locally st. (2) λ = α Cl(λ) for a fuzzy st α. (3) λ = λ Λ Cl(λ). Proof. (1) (2) : Lt λ = α β whr α is a fuzzy Λ st and β is a fuzzy -closd st. Sinc λ α and λ Cl(λ), w hav λ α Cl(λ). On th othr hand, λ β and λ Cl(λ) Cl(β) = β, α Cl(λ) λ which complts th proof. (2) (3) : If λ = α Cl(λ) for a fuzzy st α, thn λ α. Thus, λ α = α which implis λ Cl(λ) α Cl(λ) = λ. Sinc λ λ and λ Cl(λ), λ λ Cl(λ). (3) (1) : Sinc λ is a fuzzy st and Cl(λ) is a fuzzy -closd st, λ is a fuzzy locally st.

7 A. Vadivl t al. / J. Linar. Topological. Algbra. 06(02) (2017) Fuzzy continuity and Fuzzy locally continuity In this sction, w prsnt two wakr forms of fuzzy continuity namd fuzzy continuity and fuzzy locally continuity via th fuzzy sts and fuzzy locally sts and w obtain som charactrizations of ths nw continuitis. Dfinition 5.1 Lt f : (X, τ 1 ) (Y, τ 2 ) b a function from a fuzzy topological spac (X, τ 1 ) into a fuzzy topological spac (Y, τ 2 ). Th function f is calld (1) fuzzy continuous if f 1 (µ) is a fuzzy st of X for ach µ τ 2. (2) fuzzy locally continuous if f 1 (µ) is a fuzzy locally st of X for ach µ τ 2. Rmark 6 It is clar that th implications of th following diagram hold. fuzzy continuous fuzzy δ-smi continuous fuzzy δ-pr continuous fuzzy -continuous fuzzy -continuous fuzzy locally -continuous Howvr, non of th implications of this diagram is rvrsd as shown in th following xampls. Exampl 5.2 Lt X = {a, b, c} and λ, µ, γ and δ b fuzzy sts of X dfind as follows: λ(a) = 0.4, λ(b) = 0.6, λ(b) = 0.5; µ(a) = 0.6, µ(b) = 0.4, µ(c) = 0.4; γ(a) = 0.6, γ(b) = 0.4, γ(c) = 0.5; δ(a) = 0.4, δ(b) = 0.5, δ(c) = 0.5. Lt τ 1 = {0, 1, λ, µ, λ µ, λµ}, τ 2 = {0, 1, γ} and τ 3 = {0, 1, δ}ar fuzzy topologis on X. Considr th idntity mapping f : (X, τ 1 ) (Y, τ 2 ) and g : (X, τ 1 ) (Y, τ 3 )dfind by f(x) = x and g(x) = x, x X. It is clar that f is fuzzy continuous, but it is not fuzzy δ-pr continuous. Similarly, g is fuzzy -continuous, but it is not fuzzy δ-smi continuous. Exampl 5.3 Lt λ and µ b fuzzy substs of X = Y = {a, b} ar dfind as follows: λ(a) = 0.2, λ(b) = 0.3; µ(a) = 0.1, µ(b) = 0.4; Lt τ 1 = {0, 1, λ} and τ 2 = {0, 1, µ} ar fuzzy topologis on X. Considr th idntity mapping f : (X, τ 1 ) (Y, τ 2 ) dfind by f(x) = x, x X. Hr, µ is fuzzy opn st in Y, f 1 (µ) = µ is a st in X. Hnc, f is fuzzy continuous but f is not fuzzy continuous as th fuzzy st µ is fuzzy opn st in Y, but f 1 (µ) is not fuzzy -opn st in X. Thus, f is fuzzy continuous but f is not fuzzy -continuous. Exampl 5.4 Lt λ and µ b fuzzy substs of X = Y = {a, b} ar dfind as follows: λ(a) = 0.2, λ(b) = 0; µ(a) = 0.1, µ(b) = 1; Lt τ 1 = {0, 1, λ} and τ 2 = {0, 1, µ} ar fuzzy topologis on X. Considr th idntity mapping f : (X, τ 1 ) (Y, τ 2 ) dfind by f(x) = x, x X. Hr, µ is fuzzy opn st

8 132 A. Vadivl t al. / J. Linar. Topological. Algbra. 06(02) (2017) in Y, f 1 (µ) = µ is a fuzzy locally st of X. Hnc, f is fuzzy locally continuous but f is not fuzzy continuous as th fuzzy st µ is fuzzy opn st in Y, but f 1 (µ) is not fuzzy st in X. Thus, f is fuzzy locally continuous but f is not fuzzy continuous. Exampl 5.5 Lt λ, µ, γ and δ b fuzzy substs of X = {a, b, c} dfind as follows: λ(a) = 0.3, λ(b) = 0.4, λ(c) = 0.5; µ(a) = 0.6, µ(b) = 0.5, µ(c) = 0.5; γ(a) = 0.3, γ(b) = 0.5, γ(c) = 0.2; δ(a) = 0.2, δ(b) = 0.2, δ(c) = 0.2. Lt τ = {0, 1, λ, µ} and η = {0, 1, δ} ar fuzzy topologis on X. Considr th idntity mapping f : (X, τ) (Y, η) dfind by f(x) = x, x X. Hr, th idntity function f : (X, τ) (Y, η) is fuzzy -continuous but not fuzzy continuous bcaus for any δ η, f 1 (δ) / τ. Thorm 5.6 Lt f : (X, τ 1 ) (Y, τ 2 ) b a function from a fuzzy topological spac (X, τ 1 ) into a fuzzy topological spac (Y, τ 2 ). Th following statmnts ar quivalnt: (1) f is fuzzy continuous. (2) For all µ c τ 2, f 1 (µ) (X). (3) For all fuzzy st λ of Y, (f 1 (Intλ)) f 1 (λ). Proof. (1) (2) : Lt µ c τ 2. Sinc f is continuous, f 1 (µ c ) = (f 1 (µ)) c (X). Thus, f 1 (µ) (X). (2) (1) : It can b provd in th abov mannr. (1) (3) : Sinc Intλ τ 2 and f is continuous. (f 1 (Intλ)) = f 1 (Intλ) f 1 (λ). (3) (1) : Lt µ τ 2. Thn Intµ = µ. By assumption, (f 1 (µ)) = (f 1 (Intµ)) f 1 (µ). Sinc, f 1 (µ) (f 1 (µ)) always holds, f 1 (µ) (X). Thorm 5.7 Lt f : (X, τ 1 ) (Y, τ 2 ) b a function from a fuzzy topological spac (X, τ 1 ) into a fuzzy topological spac (Y, τ 2 ). Th following statmnts ar quivalnt: (1) f is fuzzy locally continuous. (2) For all fuzzy st λ of Y, f 1 (Intλ) = (f 1 (Intλ)) Cl(f 1 (Intλ)). (3) For all fuzzy st λ of Y, f 1 (Clλ) = (f 1 (Clλ)) Int(f 1 (Clλ)). Proof. (1) (2) : Sinc Intλ is a fuzzy opn st, th proof is immdiat from Thorm 4.4 (1) (3) : (Clλ) c is a fuzzy opn st, so by Thorm 4.4 f 1 ((Clλ) c ) = (f 1 (Clλ)) c = ((f 1 (Clλ)) c ) Cl((f 1 (Clλ)) c ) = ((f 1 (Clλ)) ) c (Int(f 1 (Clλ))) c,

9 A. Vadivl t al. / J. Linar. Topological. Algbra. 06(02) (2017) f 1 (Clλ) = (f 1 (Clλ)) Int(f 1 (Cl(λ)). (3) (1) : Lt λ b a fuzzy opn st. Thus, Cl(λ c ) = λ c. By assumption, f 1 (Cl(λ c )) = f 1 (λ c ) = (f 1 (Cl(λ c ))) Int(f 1 (Cl(λ c ))) = (f 1 (λ c )) Int(f 1 (λ c )) = ((f 1 (λ)) c ) Int((f 1 (λ)) c ) = ((f 1 (λ)) ) c (Cl((f 1 (λ)))) c. = (f 1 (λ)) (Cl((f 1 (λ)))) c. Hnc, f 1 (λ) = (f 1 (λ)) Cl((f 1 (λ))) which mans f 1 (λ) is a fuzzy locally st. Thorm 5.8 Lt f : X Y b a fuzzy continuous function and g : Y Z fuzzy continuous function. Thn g f : X Z is a fuzzy continuous function. Proof. It is clar from th quality (g f) 1 = f 1 (g 1 (µ)). Thorm 5.9 Lt f : X Y b a fuzzy continuous function. If g : X X Y th graph map of f is fuzzy continuous, thn f is a fuzzy continuous function. Proof. Lt µ b an opn st of Y. Thn 1 X µ is an opn st of X Y. By Lmma 2.4 in [1], g 1 (1 X µ) = 1 X f 1 (µ) = f 1 (µ) (X). Conclusion This papr dals with th rcnt concpts in th litratur. Th concpt of fuzzy sts is studid via -opn sts. So, this papr is rlatd to [7 11] in th litratur. Acknowldgmnts Th authors would lik to thank from th anonymous rviwrs for carfully rading of th manuscript and giving usful commnts, which will hlp to improv th papr. Rfrncs [1] K. K. Azad, On fuzzy smi continuity, fuzzy almost continuity and fuzzy wakly continuity, J. Math. Anal. Appl. 82 (1981), [2] F. G. Arnas, J. Dontchv, M. Ganstr, On λ sts and th dual of gnralizd continuity, Qustions and answrs in Gnral Topology. 15 (1997), 3-13.

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