Journal of Pure Mathematics, Vol-22

2005·3.4 MB·

Volume 22, 2005
DEPARTMENT OF PURE MATHEMATICS, UNIVERSITY OF CALCUTTA
35, Ballygunge Circular Road, Kolkata - 700 019, India
JOURNAL OF PURE MATHEMATICS
UNIVERSITY OF CALCUTTA
\
Board of Editors Editorial Collaborators
L. Carlitz B. Barua
W. N. Everitt B. C. Chakraborty
L. Debnath M. K. Chakraborty
T. K. Mukheijee D. K. Bhattacharya
M. C. Chaki D. K. Ganguly
N. K. Chakraborty T. K. Dutta
J. Das J. Sett
S. K. Chatterjea M. Mazumdar
M. Dutta
P. K. Sengupta (Managing Editor, Ex-officio)
S. Ganguly (Managing Editor)
M. K. Sen (Managing Editor)
S. K. Acharyya (Managing Editor)
Chief Managing Editor
M. N. Mukherjee
The Journal of Pure Mathematics publishes original research
work and/or expository articles of high quality in any branch of
Pure Mathematics. Papers (3 copies) for publication/Books for
review/Joumal in exchange should be sent to :
A 6 0°>0
<3i* l
The Chief Managing Editor
Journal of Pure Mathematics
Department of Pure Mathematics
University of Calcutta
35, Ballygunge Circular Road
Kolkata 700 019, India
A NEW CLASS OF FUNCTIONS
A. AgiKGOZ And S. YCksel
ABSTRACT : In this paper, we introduce a new class of functions called strongly pre - / - continuous
functions in ideal topological spaces. Several characterizations and some of its properties of this type
of functions are obtained. We also investigate the relationship between this class of functions and other
classes of non-continuous functions.
Key words and pharases : P - / - open, semi - 7 - open, pre - / - open, P - 1 - open set and strongly
pre - / - continuous function.
2000 Mathematics Subject Classification Codes : 54C08.
1. INTRODUCTION
The purpose of the present paper is to introduce and investigate the notion of a new
class of functions, namely strongly pre - 7 - continuous functions, and give several
characterizations and their properties. Relationships between this type of functions and other
classes of functions are obtained. This new class of strongly pre - 7 - continuous functions,
which is stronger than a - pre - 7 - continuous [20], is a generalization of strongly a - /
- continuous [18].
For the understanding of this paper, we should recall nine classes of functions : strongly
a - I - continuous [18], almost - 7 - irresolute [19], a - I - irresolute [20], P - / - irresolute
[19], a - pre - 7 - continuous [20], almost a - I - irresolute [20], a - I - condnuous [8],
pre - / - continuous [6] and P - / - condnuous [8] functions in ideal topological spaces.
2. PRELIMINARIES
Throughout this paper Cl (A) and Int (A) denote the closure and the interior of A,
respectively. Let (X, x) be a topological space and 7 be an ideal of subsets of X. An ideal,
I is defined as a nonempty collection of subsets of X satisfying the following two conditions:
(1) If A e 7 and B c A, then B e 7; (2) If A e 7 and B e 7, then A u B e 7. An ideal
topological space is a topological space (X, x) with an ideal 7 on X and is denoted by (X,
x, 7). For a subset A c X, A*(7) = [xeX | UnAg7 for each neighborhood U of x) is called
the local function of A with respect to 7 and x [II]. We simply write A* instead of A*(7)
2 A. AQIKGOZ AND S. YOKSEL
since there is no chance for confusion. X* is often a proper subset of X. The hypothesis X=X*
[9] is equivalent to the hypothesis x n /= 0 [17]. For every ideal topological space (X, x,
/), there exists a topology x*(/), finer than x, generated by P (I, x) = {U \ 1 : Uex and Ig/},
but in general P (/, x) is not always a topology [10].
Additionally, C1*(A) = AuA* defines a Kuratowski closure operator fa x*(/).
We recall some known definitions. ;
Definition 2.1. A subset S of a topological space (X, x) is said to be a - open [14]
(resp. semi - open [ 12], pre - open [13],/?-open [ 1 ]) if S c Int(Cl(Tnt(S))) (resp. S c Cl(Int(S)>,
S c Int(Cl(S)), S c CI(Int(Cl(S)))).
The family of all a - open (resp. semi - open, pre - open, P - open) sets in a space
X is denoted by x“ (resp. SO(X), PO(X), PO(X)). Itjis shown in [14] that x“ is is a topology
for X and x c x“ c SO(X). Moreover, it is shown in [18] that SO(X) u PO(X) c P0(X).
The complement of a semi - open (resp. pre - open) set is semi - closed [4] (resp. pre -
closed [13]). The interesectidn of all semi - closed sets containing S is called the semi - closure
[4] of S and is denoted by sCI(S); the union of all semi - open sets contained in S is called
the semi - interior [4] of S and is denoted by slnt(S).
Definition 22. A subset A of an ideal topological space (X, x, /) is said to be a - l
- open [7] (resp. semi -I - open [7], p re -l- open [6], f$ - / - open [7] if A c Int(Cl*(Int(A)))
(resp. A c Cl*(lnt(A), A c lnt(CI*(A)), A c Cl(lnt(Cl*(A)))). The family of all a - l - open
(resp. semi - / - open, pre - / - open, P - / - open) sets in an ideal topological space (X,
x, I) is denoted by a/0(X) (resp. SZO(X), P/0(X),; p/0(X)).
Definition 23. A function f : (X, x, /) —> (Y, t>) is called a - I - continuous [7] (resp.
pre - I - continuous [6], /J - / - continuous [7] if for every V 6 i), f-1 (V) is a - I - open
(resp. pre - / - open, P - / - open) in (X, x, /). !
Definition 2.4. A function f : (X, x, /) —> (Yj v) is said to be a - I - irresolute [20]
(resp. a - pre - I - continuous [20], cdmost a - I - irresolute [20]) if f-1 (V) is a - I -
open (resp. pre - / - open, P - / - open) in X for every a - open set V of Y.
Definition 2.5. A function f : (X, x, I) (YJ o) is said to be fi - I - irresolute [19]
(resp.. strongly a - pre - I - irresolute, strongly fi - pre - I - irresolute) if (V) is p -
I - open (resp. a - / - open, pre - / - open) in X for every p - open set V of Y.
Definition 2.6. A function f : (X, x, I) —> (Y, n) is said to be strongly a-1 - continuous
[18] (resp. strongly pre - I - continuous, almost I - irresolute [19]) if f-1 (V) is a - / - open
(resp. pre - / - open, P - / - open) in X for every semi - open set V of Y.
A NEW CLASS OF FUNCTIONS 3
Using the definitions above, we obtain the following diagram:
strong a - pre - / - irresoluteness —> strong a - / - continuity a - / - irresluteness —MX - / - continuity
J, J,
strong (3-pre-/ - irresoluteness —) strongly pre-/-continuity —> a -pre -/-continuity pre -/-continuity
'i' Jr J' Jr
P~/~ irresoluteness —^almost-/- irresoluteness —> almost a -/- irresolute —> (3-/- continuity
The following remark • enables us to realize that none of the above implications is
reversible.
Remark 2.7. We have the following relationships:
(a) a - I - irresoluteness does not imply almost I - irresoluteness ([19, Example 1.2]),
(b) a - I - continuity does not imply almost a - I - irresoluteness ([19, Example 1.3])
(c) Strong (3 - pre - / - irresoluteness does not imply a - I - continuity ([20, Example
1.1]).
(d) (3 - / - irresoluteness does not imply pre - I - continuity ([20, Example 1.2]).
3. STRONGLY PRE - / - CONTINUOUS FUNCTIONS
Theorem 3.1. For a function f : (X, T, I) —> (Y, o), the following are equivalent:
(i) f is strongly pre - / - continuous;
(ii) For each x G X and each semi - open set V of Y containing f(x), there exists
a pre -l - open set U of X containing x such that f(U) c V;
(Hi) f“'(V) c InUGl*^1 (V))) for every semi - open set V of Y;
. (iv) f-I(P) is pre - 1 - closed in X for every semi - closed set F of Y;
(v) Cl(Int*(f-1(B))) c f-'(sCl(B)) for every subset B of Y;
(vi) f(Cl(int*(A))) c sCl(f(A)) for every subset A of X.
Proof, (i) => (ii). Let x G X and let V be any semi - open set of Y containing f(x).
By Definition 2.6, f”*(V) is pre - I - open in X and contains x. Set U = f^fV), then by (i),
U is a pre - I - open subset of X containing x and f(U)- a V.
(ii) => (Hi). Let V be any semi - open set of Y and x G f^1 (V). By (ii), there exists
a pre -l - open set U of X containing x such that fiTJ) c V. Thus, we have
xGUcInt(Cl*(U))c!nt(Cl*(fM(V))) and hence f-*(V) c Int(Cl*(f-1(V))).
4 A. AgiKGOZ AND S. YOKSEL
(Hi) => (iv). Let F be any semi - closed set of Y. Set V = Y - F, then V is semi -
open in Y. By (iii), H(V) c Int(Cl*(r‘(V))) and hence r‘(F) = X - f-‘(Y - F) = X -
f-‘(V) is pre - I - closed in X.
(iv) => (v). Let B be any subset of Y. Since sCl(B) is a semi - closed subset of Y, by
(iv) f^sCKB)) is pre - / - closed and X - f“,(sCl(B)> is pre - / - open. Thus X - H(sCI(B))
c Int(Cl*(X - r^sCKB)))) = Int(X - Int^f-'CsCKB)))) = X - Cl(Int*(r>(sCI(B)))). Therefore,
we obtain Cl(lnt*(r'(B))) c f-'(sCl(B)). . ,
(v) => (vi). Let A be any subset of X. By (v), we have Cl(Int*(A)) c CKlnt^r^flA))))
c f‘1(sCl(f(A))) and hence f(Cl(Int*(A))) c sCI(f(A)).
!
(vi) => (i). Let V be any semi - open of Y Then, by (vi), f(Cl(Int*(f-'(Y - V)))) c
sCl(f(f-‘(Y - V))) c sCl(f(f-‘(Y - V))) c sCl(Y - V) = Y - slnt(V) = Y - V. Thus, we have
Ciant^CY - V))) c r'(Y - V) and hence X - IritCCl^r'CV))) = Cl(Int*(f-‘(Y - V))) c
X - f-'CV). Consequently, we obtain f-"(V) c IntfCl^f-'CV))). This shows that r‘(V) is pre
-I - open. Therefore, f is strongly pre - I - continuous.
|
Lemma 3.2. ([15], [7]). Let [X^ ; XeA] be a family of spaces and be a nonempty
subset of X;y for each i = 1,2, ..., n. Then U = I\ ^ xi x H, = n is a nonempty semi
- open [15] (resp. pre - open [7]) subset of FI X^ if and only if is semi - open (resp.
pre - open) in Xw for each i = I, 2, .... n.
Theorem 3.3. A function f : (X, x, /) —> (Y "u) is strongly pre - / - continuous if the
graph function g : (X, x, /) —> X x Y, defined by g(x) = (x, f(x)) for each x 6 X, is strongly
pre - / - continuous.
Proof. Let x 6 X and V be any semi - open set of Y containing f(x). Then, X x V
is a semi - open set of X x- Y by Lemma 3.2 and contains g(x). Since g is strongly pre -
I - continuous, there exists a pre - / - open set Ui of X containing x such that g(U) c X
x V and hence f(U) c V. Thus f is strongly pre -1 / - continuous.
Theorem 3.4. If a function f : (X, x, /) —> flY^ is strongly pre - / - continuous, then
of: (X, x, I) —> is strongly pre - / - continuous for each X e A, where Px is the projection
of UYX onto Y^.
I
Proof. Let be any semi - open set of Y^. Since P^ is continuous and open, it is
irresolute by Theorem 1.2 of [5] and hence PX-1(VX) [s semi - open in IlY^. Since f is strongly
pre - / - continuous, we obtain rl(P^_1(V^)) = (P^ of)_1(V^) and is pre - open in X. This
shows that P^ of is strongly pre - / - continuous, for each X e A.
Lemma 3.5. ([2]) Let (X, x, f) be an ideal topological space and A be a subset of X.
Then the following properties hold: ,
(1) -If O is open in (X, x, I), then O n C1*(A) c Cl*(OnA).
(2) If A c X0 c X, then Cl* x0 (A) = C1*(A) n X0.
A NEW CLASS OF FUNCTIONS 5
Lemma 3.6. Let (X, x, f) be an ideal topological space. If AeP/O(X0) and
AcXqG PZO(X), then AgP/0(X).
Proof. Since AeP/O(X0), A c (A)) = X0 n U for some Uex. Since
X0eP/O(X), by Lemma 3.5 (2),
AcXjnU cU n ImfC!*^)
= Int(U n C1*(X0))
c Int(Cl*(U n Xy))
= Int(Cl(A))))
= Int(Cl*(lnt(C/*Vj (A) n Xq)))
c Int(C(A) n X0))
= Int(Cl*(Cl*(A) n X0))
c Int(Cl*(Cl*(A)))
c lnt(Cl*(A)).
This shows that AeP/0(X).
Lemma 3.7. Let (X, x, I) be an ideal topological space. If AgP/(X) and X0eS/O(X),
then AnX06P/O(X0).
Proof. AnX0clnt(Cl<‘(A))nX0 = Int^ ant(Cl*(A))nX0)
c lmXQ [Int(CI*(A))nCl*(Int(X0))] c IntXQ [Cl*(Int(Cl*(A))nlnt(X0))]
c lntXQ [Cl*(Cl*(A)nInt(X0))]
c /nL0[CU(Cl*(AnX0))]
c [Cl*((AnX0)]
c Int%0[C\*{AnXQ)]nX0
c /«tXo[Cl*(AnX0)]nc IntXo(X0)
c/nrXo[Cl*(AnX0)nX0]
cz Int^iCrXniA'XJ).
6 A. AQ1KG0Z AND S. VOKSEL
I
This shows that AnX0e P70(Xg). '
Theorem 3.8. If f ; (X, x, 7) —> (Y, o) is strongly pre ~ / - continuous and A is a semi
- 7 - open subset of X, then the restriction f / A : A| —> Y is strongly pre -l - continuous.
I
Proof. Let V be any semi - open set of Y. Since f is strongly pre - 7 - continuous,
then f"1(V) is pre - 7 - open in X. Since A is semi - 7 - open in X, by Lemma 3.4,
(f/A)_1(V) = Anr‘(V) is pre - 7 - open in A. Hence f/A is strongly pre - 7 - continuous.
Theorem 3.9. Let f : (X, x, 7) —> (Y, o) be a function and {A^ : XeA} be a cover
of X by pre ~ 7 - open sets of (X, x, 7). Then f is strongly pre - 7 - continuous if HAX :
A^ —> Y is strongly pre - 7 - continuous for each XeA.
Proof. Let V be any semi - open set of Y. Since f/A^ is strongly pre - 7 - continuous,
then (f/AjJr'CV) is pre - 7 - open in Ax. Since Ax is pre - 7 - open in X, by Lemma 3.6,
(lyAj^CV) is pre - 7 - open in X for each XeA. Therefore, f_1(V) = Xn f"'(V) = u{ A^n
f^O/) : Xe A} = u{(f/A^)_l(V) : X gA) is pre - 7 - open in X because the union pre -
7 - open sets is a pre - 7 - open set [6]. Hence f is strongly pre - 7 - continuous.
I
Definition 3,10. [5] A function f : (X, x, I) —>i (Y, o) is said to be irresolute if f-'CV)
is semi - open in X for every semi - open set V df Y.
Theorem 3.11. Let f : (X, x, 7) —> (Y, o) —> Z be functions, then the composition g
of : (X, x, 7) —> Z is strongly pre - I - continuous if f is strongly pre - 7 - continuous and
g is irresolute.
Proof. Let W be any semi - open subset of Z. Since g is irresolute, then g~*(W) is
semi - open in Y. Since f is strongly pre - 7 - continuous, then (gofr'(W) = L'(g_1(W))
is pre - 7 - open in X and hence gof is strongly pre - 7 - continuous.
Definition 3.12. [3] A space (X, x, 7) is said to be 7 - submaxinial if every x* - dense
subset of X is open in X. '
Definition 3.13. [3] A space (X, x, 7) is said to be P - I - disconnected (briefly P.I.d.)
if the 0 * A* e x for each Aex. 1
Theorem 3.14. Let (X, x, /) be an 7 - submaximal and P - 7 - disconnected space.
Then, the following are equivalent for a function f : (X, x, 7) —> (Y, d) :
(i) Strongly pre - 7 - continuity <=> strong| a - 7 - continuity.
(ii) Strong a - pre - 7 - irresoluteness «: strong p - pre - 7 - irresoluteness.
(Hi) a - I - irresoluteness <=> a - pre - 7 J- continuity.
(iv) Continuity <=> a - / - continuity « pre - 7 - continuity.
A NEW CLASS OF FUNCTIONS 7
Theorem 3.15. Let (X, x. T) be a / - submaximal and P - / - disconnected space. Then,
the following are equivalent for a function f : (X, x, T) —> (Y, t») :
(i) Strongly pre - / - continuity <=> a - pre - / - continuity o pre - / - continuity.
(U) Strong a - / - continuity <=> a - I - irresoluteness o a - / - continuity.
Proof of Theorem 3.14 and 3.15 follows from the fact that if (X, t, /) is / - submaximal
and P - / - disconnected space, than x = o/0(X) = S/0(X) = P/0(X) [3].
The authors wish to thank the referee for this valuable suggestions.
REFERENCES
1. M.E. Abd E!-Monsef, S. N. El - Deep and R. A. Mahmoud, P - open sets and (3 - continuous
mapping. Bull. Fac. Sci. Assiut Univ., 12 (1983), 77 - 90.
2. A. Acikgoz, T. Noiri and S. Yuksel, On a - / - continuous and a - I - open functions, Acts
Math. Hungar., 105 (1 - 2) (2004), 27 - 37.
3. S. Yuksel, A. Acikgoz and T. Noiri, a - I - preirresolute functions and P - / - preirresolute
functions. Bull. Malaysian Math. Sci. Soc., Vd 28 (i) (2005).
4. S. G. Crossley and S. K. Hildebrand, Semi - closure, Texas J. Sci., 22 (1971), 99 - 112.
5. S. G. Crossley and S. K. Hildebrand, Semi - topological properties. Fund. Math., 74 (1972),
233 - 254.
6. J. Dontchev, On pre - / - open sets and a decomposition of / - continuity. Banyan Math.
J., 2 (1996).
7. N. El - Deeb, 1. A. Hasanein, A. S. Mashhour and T. Noiri, On p - regular spaces. Bull.
Math. Soc. Sci. Math. R. S. Roumanie (N. S.), 27 (75) (1983), 311 - 315.
8. E. Hatir and T. Noiri, On decompositions of continuity via idealization, Acta Math. Hungar.,
96 (4) (2002), 341 - 349.
9. E. Hayashi, Topologies defined by local properties, Math. Ann., 156 (1964), 205 - 215.
10. D. Jankovic and T. R. Hamlett, New topologies from old via ideals, Amer. Math. Monthly,
97 (1990), 295 - 310.
11. K. Kuratowski, Topology, Vol. 1 (transl.). Academic Press. New York, 1966.
12. N. Levine, Semi - open sets and semi - continuity in topological spaces, Amer. Math. Monthly,
70 (1963), 36 - 41.
13. A. S. Mashhour, M. E. Abd El - Monsef and S. N. El - Deep, On precontinuous and weak
precontinuous mappings, Proc. Math. Phys. Soc. Egypt, 53 (1982), 47 - 53.
8 A. AgiKGOZ AND S. YOKSEL
t
14. 0. NjSstad, On some classes of nearly of open I sets Pacific J. Math., IS (3) (1965), 961 -
' ’ ' j
970. "
15. T. Noiri, Remarks on semi - open mappings, Bull. Calcutta Math. Soc., 65 (1973), 197 -
i
201.
1(
16. T. Noiri and V. Popa, Weak forms of faint continuity, Bull. Math. Soc. Sci. Math R. S. Roumanie
(N. S.) 34 (82) (1990). j
17. P. Samuels, A topology formed from a given topology and ideal, J. London Math. Soc., (2)
10 (1975), 409 - 416. I
I
18. S. Yuksel, A. Acikgoz and T. Noiri, On strongly a - l - continuous functions, Far East Journal
Sci., 1 (2003) 1 - 18. ;
i
19. S. Yuksel and A. Acikgoz, On almost a - l - irresolute functions, (submitted).
i
20. S. Yuksel, A. Acikgoz and T. Noiri, On a - pre |- / - continuous functions. News Bull. Cal.
Math. Soc., 26 (7 - 9) (2003) 7 - 12. !
Selcuk University
Department of Mathematics
42031 Campus-Konya
Dirkey
E-mail: [email protected]
I
!
i