对一些弱连续性的讨论

总 39卷 第2期 2006年6月 数 学 研 究 Journal of Mathematical Study Vo1．39 No．2 Jun．2006 On Some W eak Forms of Continuity }rang Erguang (Department of M athematics。Anhui University，Hefei 230039) Abstract In this paper．the notions of weakly semi-continuous functions and weakly pre—con- tinuous functions are introduced，and some of their properties are investigated．Also ，we investigate the interrelations between some weak forms of continuity． Key words Weak口一continuity；weak semi-continuity；weak pre-continuity CLC humber O189．1 Document code A 1 IntrOductiOn In the past several years，some weak forms of continuity were introduced to generalize conti— nuity．Levine[13 defined weakly continuous functions as a generalization of continuity．In[2]， Noiri introd uced the notion of weak a-continuity to generalize weak continuity．In this paper，we introduce tWO classes of functions．namely weakly semi— continuous functions and weakly pre— continuous functions．Both of them are generalizations of weak a-continuity．Some of their prop- erties are investigated in section 2． In section 3，we make some comparisons between certain weak forms of continuity． Throughout，a space will mean a topological space．Let A be a subset of a space( ， )． We denote the closure and the interior of A by el(A)and int(A)，respectively． A subset A of a space(x，3-)is called semi—open[33(resp．pre—open[4]，regular open [s]，an a-set[6])if ACcl(int(A))(resp．ACint(cl(A))，A=int(cl(A))，ACint(cl(int ( ))))．The family of all semi—open (resp．pre—open，regular open)sets and口一sets of a space ( ，3-)will be denoted by SO( ， )(resp．P0( ，9-)，RO(X，9-))and 3-。．A iS said to be semi—closed(resp．pre—closed，regular closed)if — is semi—open(resp．pre—open。regular open)．The semi-closure(resp．pre-closure)of a subset A，written scl(A)(resp．pcl(A))，is the intersection of all semi-closed (resp．pre-closed)subsets of(X， )that contain A．The se— mi-interior(resp．pre-interior)of A，denoted by sint( )(resp．pint(A))。is the union of all semi—open(resp．pre-open)subsets of( ， )contained in A． Proposition I．I[’] Let A and B be subsets of a space( 。3-) ． Then Received date：2004— 11—16 Foundation item：This work is supported by NNSF of China(10071018)and the EYTP of China 维普资讯 http://www.cqvip.com ·146· 数学研究 2006正 (a)If A∈SO(X，3-)and B∈P0( ，9-)，then AnB∈SO(B)； (b)If A∈S0(B)and B∈S0(X， )，then A∈S0(X ，9-)． Proposition 1．2 Let A be a subset of a space( ， )，then (a)sint(A)= Ncl(int(A))； (b)scl(A)=AUint(cl( ))． Definition 1．3 A function f：( ， ) (y， )is said to be (a)semi—continuous[3](resp．pre—continuous[4f1)if f一 ( )∈SO( ， )(resp．f—l ( )∈P0( ， ))for each ∈ ； (b)weakly a-continuous[2](briefly，w．口．c)(resp．weakly continuous[1])if for each z ∈X and each V∈ containing厂(z)，there exists U∈ (resp． ∈9-)containing z，such that厂( )Ccl( )； (c)almost semi—continuous[9](briefly，a．s．c．)(resp．almost pre—continuous[1 0] (briefly，a．p．c．))if for each z∈X and each VffRO(Y， )containing厂(z)，there exists UE S0(X，9-)(resp．U∈PO(X， ))containing z，such that厂( )CV； (d)almost weakly continuous[11]if f一’(V)Cint(cl(f一 (c ( ))))for each Vff ； (e)weakly quasi continuous[12] f for each z∈X and each G∈ containing z and each V ∈ containing厂(z)，there exists ∈ such that j2『≠UCG and f(U)Ccl(V)． Definition 1．4 A function f：(X， ) (y， )is said to be weakly semi—continuous (briefly，w．s．c．)(resp．weakly pre=continuous(briefly，w．P．c．))if for each z∈X and each VE containing厂(z)，there exists U∈SO(X， )(resp．U∈P0(X， ))containing z， such that厂(U)Ccl( )． 2 Properties of weak semi—continuity and weak pre-continuity In this section，we investigate the properties of w．s．c．functions and w．P．c．functions．To simplify our narration，we only state the properties of w．s．c．functions．The properties of w．P． c．functions can be obtained by slightly modifying those of w．s．c．functions． Theorem 2．1 For a function f：(X， ) (y，aed)，the following are equivalent (a)f is w．s．c．； (b)f一 ( )Ccl(int(f一 (c ( ))))for each Vff ； (c)int(cl(f ( )))Of_1(cf( ))for each V∈ ； (d)scl(f一 ( ))C广 (c ( ))for each V∈ ； (e)广 (V)Csint(f_1(cf(V)))for each V∈ ． Proof (a) (b) Let V6 ．For every z∈f_1( )，we have厂(z)∈V．Sincefis w．s． c．，there exists U∈SO(X， )containing工such that厂(【，)Ccl( )．That is，UCf_1(cl ( ))．Hence UCcl(int( ))Ccl(int(f_1(c ( ))))，which implies that f (V)Ccl(int(f (cl ( ))))． (b) (c) For every Vff ，we have Y＼c ( )∈ ，and so f_1(y＼c ( ))Ccl(int(厂_1 (c (y＼cl( )))))Ccl(int(厂 (cl(y＼ )))= cl( 2￡(厂_1(y＼ )))一 ＼int．(cl(厂_1( )))． Therefore，int(cl(：f一 ( )))Cf一 (c (V))． 维普资讯 http://www.cqvip.com 第2期 Yang Erguang：On Some Weak Forms of Continuity ·l47· (c) (d) This can be seen from Proposition 1．2(b)，since for each V∈ ，we have厂 ( )c， ( (V))． (d) (e) Suppose that scl(f-1( ))cf一 ( ( ))for each V∈ ．Then，by Proposition 1．2(b)，int(cl(f ( )))cf ( ( ))．For every V6 ，we have y＼ ( )∈ ，so X＼cl (int(广 (cl( ))))一 int(cl(厂 (y ( )))) 广 (cl(y＼ ( )))cf-1(y＼ )一 ＼广 ( )．Therefore，厂 ( )c (int(厂-1(cl( ))))．By Proposition 1．2(a)，one easily verifies ‘hat(e)holds． (e) (a) Suppose that(e)holds．Then for each ∈X and each V∈ containing厂 ( )，we have ∈厂 ( )，and so ∈sint(广 ( ( )))．Let U=sint(广 ( ( )))．Then ∈ SO(X， )containing and UCf ( ( ))．Hence厂( ) ( )，whichimplies that厂is w． S C． Theorem 2．2 If厂：( ， )一 (y， )is w．s．c．and A∈P0( ， )，then the restriction 厂IA：( ， I )一(y， )is w．s．c． Proof For each ∈ACX and each V6 containing(厂I )( )一厂( )，since厂is w．s． c．，there exists U∈SO(X，3-)containing such that f(U)Ccl(V)．Put W=UNA，since A ∈PO( ， )，by proposition 1．1(a)，we have W ∈SO( ， I )containing and(厂IA) ( )一厂( )c厂( ) ( )．This indicates that厂I is w．s．c． Theorem 2．3 Let厂：( ， )一 (y， )be a function and{U，：5∈S)a cover of X by se— mi—open sets of(X， )．If厂I【，，：(【，，， I【，，)一 (y， )is w．s-c．for each s∈S，then厂is W ．S．C． Proof For each ∈X，there exists 5o∈S such that ∈U Since厂IU is w．s．c．，for each V6 containing(fl ，。)( )=厂( )，there exists W ∈SO(U,。， I ，。)containing z such that厂( ，。)一(厂I 。)( ，。)CcI(V)．Since U ∈SO(x，9-)，by Proposition 1．1(b)， ∈ O( ， )．Therefore，f is w．s．c． Theorem 2．4 A function厂：( ， ) (y， )is w．s．c．if and only if the graph function g：( ， )一 ( ×y， × )defined by g( )一(z，厂( ))for every ∈X is w．s．c． Proof Suppose that厂is w．s．c．Let ∈X and g(z)∈WE9-× ，then there exist Ul∈ and V∈ such that( ，厂( ))∈ l× ．Since厂is w．s．c．，there exists U2∈ O( ， )containing such that f(U2)c ( )．Put U=UlnU2，we have ∈U∈ O( ， )and g ( )c l×f(U2)c l× ( ) ( )．Therefore，g is w．s．c． Conversely，suppose that g is w．s．c．Let ∈X and厂( )∈V6 ，then we have g( )= ( ，，( ))∈X~VE~-× ．As g is w．s．c．，there exists U6SO(X，． )containing such that g(U)Ccl(XXV)一X× ( )．Thus f(U)Ccl(V)，which implies that厂is w．s．c． 3 Comparisons In this section，vce investigate the interrelations between the weak forms of continuity previ s^ y stated． 维普资讯 http://www.cqvip.com 数学研究 2006矩 Theorem 3．1 Let f：(X， )一 (y， )and g：(y， )一 (Z， )be functions．Then the composition g。f：(X， ) (z， )is w．s．c．if f and g satisfy one of the following condi— tions (a)f is semi—continuous and g is weakly continuous； (b)fis w．s．c．and g is continuous． Proof We only prove the case of(a)．(b)can be proved analogously． For each xEX，let = (z)and z=g( )=g(，(z))．For each V∈ containing ，since g is weakly continuous，there exists UE containing y such that g( )Ccl(V)．Thus f ( ) ∈S0(X， )，because fis semi—continuous．Let U ----f ( ) then W ∈SO(X， )contain— ing and(g。，)( )=g(f(W))Cg(U)Ccl(V)．Therefore，g。f is w．s．c． Lemma 3．2[ For a function f：(X， ) (y， )，the following are equivalent (a)f is w．口．c．； ’ ‘ (b)f (V)Cint(cl(int(f (cl(V)))))for each V∈ ； (c)cl(int(cl(f_1( ))))CT_f ( ( ))for each V∈ ． Since ‘CSO(X， )and 。CPO(X， )，by their definitions，it is obvious that weak a-continuity implies weak semi-continuity and weak pre—continuity．However，the converse is not true in general as shown by the following examples． Example 3．3 Let X==：{a，b，c)， = { ，{a)，{c)，{a，c)，X)and ={(z)，{a)， {b，c)，X)．One easily verifies that the identity function f：( ， )一(X， )is weakly semi— continuous but not weakly a-continuous，since there exists(b，c)∈ such that f1((b，c))一 {b，f)is not contained in int(cl(int(厂 ( ({b，c))))))={c)． Example 3．4 Let x={口，b，c， )， 一{ ，(c)，{口， )，{口，c， )，x)and = ， {a)，{b，c)，{a，b，c)， )．Let f：( ， )一(X， )be a function defined as follows：f (口)=，(6)一，(c)=口and，( )=6．Then f is w．P．c．but not w．口．c．since there exists{a)∈ such that f ({a))={a，b，c)is not contained in int(cl(int(广 (cl({a))))))={c)． Lemma 3．5 Let A be an open subset of a space(X， )．Then cl(A)is regular closed． Proof It is dear that cl(int(c (A)))C (A)．So we need only to show that cl( )C (int( (A)))also holds．Since A is an open set，from ACcl( )，we have AC (cl(A))． Then，clearly，cl(A)Ccl(int(cl(A)))． Lemma 3．6c 。 For a function f：(X， ) (y， )，the following are equivalent (a)f is a．p．c．； (b)广 (F)is pre—closed in(X， )for every regular closed subset F of(y， )； (c)f一 ( )is pre—open in(X， )for every regular open subset V of(y， )． Theorem 3．7 For a function f：( ， )一(y， )，if f is both w．s．c．and a．P．c．，then ，is w．口．c． Proof For each V∈ ，cl( )is regular closed by Lemma 3．5．Since f is a．P．c．，by Lemma 3．6，广 ( ( ))is pre—closed in(X，9-)．Hence，cl(int(f-1( ( ))))Cf ( ( ))． Since f is also w．s．c．，by Theorem 2．1，we have int(cl(f1( )))c厂1( ( ))，which shows 维普资讯 http://www.cqvip.com 第2期 Yang Erguang：On Some Weak Forms of Continuity ·l49· that cl(int(cl(f一 ( ))))Ccl(int(f一 ( ( ))))．Hence cl(int(cl(f ( ))))cf ( ( ))． So，by Lemma 3．2，厂is w．口．c． It is easy to show that the function in Example 3．4 is a．P．c．，but，as we have already seen， it is not w．口．c．The following example shows that weak a-continuity does not imply almost pre— continuity in gener~，too． Example 3．8 Let X；{a，b，c!d}and ；{(2j，{b}，{c}，{b，c}，{a，b}，{a，b，c}， {b，c，d}， }．Let厂：(X， )一 (X，9-)be a function defined as follows：厂(口)=c，厂(6)= ， 厂(c)=6 and厂( )=口．One easny verifies that f is w．口．c．But it is neither a．P．c．nor a．s．c．， since there exists{c}∈RO(X，3-)such that，-1({c})一{a}neither belongs to P0(X， )nor belongs to 0(X， ) Theorem 3．9 For a function f：(X， ) (y， )，if厂is a．P．c．(resp．a．s．c．)，then厂 is w．P．c．(resp．w s．c． ) Proof That the theorem holds for an a．P．c．function f can be seen from Lemma 3．5， Lemma 3 6 and Theorem 2．1．Analogously，we can prove that the theorem also holds when l is a S． ． Generally speaking，weak pre—continuity (resp．weak semi—continuity)does not imply al— most pre—continuity(resp．almost semi—continuity)．This can be seen from Example 3．8．The function in Example 3．8 is w．口．c．Hence it is both w．s．c．and w．P．c．But，as we have already seen，it is neither a．s．c．nor a。P．c． Lemma 3．10c。 A function-，-：( ， ) (y， )is weakly quasi--continuous if and only if广 (V)Ccl(int(f ( ( ))))for each V∈ ． Theorem 3．11 A function f：(X， ) (y， )is w．s．c．(resp．w．P．c．)if and only if f is weakly quasi--continuous(resp almost weakly continuous)． Proof This follows from Lemma 3．10，Definition 1．3(d)and Theorem 2．1． Acknowledgment The author would like to thank his tutor professor Yan Pengfei for his valuable suggestions． References [13 Levine N．A decomposition of continuity in topological spaces，Amer．Math．Monthly，1961，68：44— 46． [23 Noiri T．Weakly a-continuous functionsf Internat．J．Math．Math．sci．，1987，10：483--490． [33 Levine N．Semi—preopen sets and semi-continuity in topological spaces，Amer．Math．Monthly，1963， 70：36— 41． [43 Mashour A S。Hasanein I A and El·Deeb S N．a-continuous and口·open mappings，Acta Math．Hungar．， 1983，41：213— 218． [53 Dugundji J．Topology．Allyn and Bacon(Boston，1972)． [63 Niastad O．On some classes of nearly open sets，Pacific J．Math．，1965，15：961—970． [73 Mashour A S，Hasanein I A and El·Deeb S N．A note on semi·continuity and pre-continuity，Indian J． 维普资讯 http://www.cqvip.com · 150- 数学研究 2006矩 [8] [9] [10] [11] [12] Pure Appl·Math．，1982．13：1119— 1123 ． Andrijevic D·Semi Preopen sets，Math ． Vesnik，1986．38：24— 32 ． 雌 san D S·Alm0stSe mi-cont．nu0us mappings ， Math．Student’l981，49：239-- 248 ． Na ef A A and No。r T·Some weak forms 0f almost c。ntinuity，Acta Math．Hunga r’'1997,74：211一 上 ． Jankovic D S· regu1ar sPaces，Intel'nat ． J．Math．Math．Sci．，1985，8：615—619 ． Popa rand Stan G()n a dec。mpos．tIon 0f quasi--continuIty in t。p。l0g．cal spaces(R。manian)，stud． 摘 要 进行了探讨． 关键词 对一些弱连续性的讨论 杨 二 光 (安徽大学数学系，安徽 合肥 230~39) 引入了弱半连续及弱准连续性的概念，讨论了他们的 一 些性质，并对某些弱连续性之间的关系 弱 a一连续 ；弱半连续 ；弱准连续 维普资讯 http://www.cqvip.com