Download bc-continuous function

Journal of Babylon University/Pure and Applied Sciences/ No.(6)/ Vol.(21): 2013
bc-continuous function
Neeran Tahir Al-Khafagy , Hanan Ali Hussein
Dept. of Math. / College of Education for Girls / Kufa University
Abstract
The main aim of this paper to study new class of continuous function is called bc-continuous
function, For this aim , the nation of b-open and α-open sets and b-compact space introduced, and we
shall study the relationship between bc-continuous and αc-continuous .
‫الخالصة‬
‫ والجل ذلك فاننا قدمنا‬bc- ‫الهدف االساسي من هذا العمل هو دراسة نوع جديد من الدوال المستمرة والتي تسمى الدوال المستمرة‬
bc - ‫ وكذلك درسنا العالقة بين الدوال المستمرة‬b - ‫ والفضاء المرصوص‬α - ‫ والمجموعة المفتوحة‬b - ‫تعريف المجموعة المفتوحة‬
. αc - ‫والدوال المستمرة‬
Introduction
Generalized open sets play a very important role in General Topology and they
are now the research topics of many topologists worldwide . Indeed a significant
theme in General Topology and Real analysis concerns the various modified forms of
continuity , separation axioms etc. by utilizing generalized open sets.D. Andrijevic
introduce a class of generalized open sets in a topological space , so called b- open
sets .The class of b- open sets is contained in the class of semi open sets and pre-open
sets.This paper includes two sections . In the first section we have dealt with the
concepts " b- compact set " ," α-compact set " and " αc- continuous function " . In
the second section , we have dealt with " bc-continuous function" , " C1 –space " and
some theorems related to these .
1. Basic Definitions
1.1 Definition [Al Omari A. and M.S.Noorani, D.Andrijevic,2008]
0
0
A subset A of a topological space X is called b-open set if A  A A .The
complement of b-open set is defined to b-closed set .
1.2 Remark [Gaber,Sama Kadhim,2010]
Every open set is b-open , but the converse is not true in general .
1.3 Definition [ Njastad O.,1965]
0
0
A subset A of a topological space X is called α-open set if A  A .The complement
of α-open set is defined to α-closed set .
1.4 Theorem [ T.Noiri,Al-Omari A. and M.S.M Noorani ,2009]
Every α-open set is b-open set in any topological space X .
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1.5 Definition [K.R.Geentry and H.B.Hoyle.III,1970]
A topological space X is called C1  space if and only if every infinite subsets of
X has a non-empty interior .
1.6 Definition [Gaber,Sama Kadhim,2010]
A subset A of a topological space X is called b- compact set relative to X , if every
cover of A by b- open sets of X has a finite sub cover .
1.7 Theorem
Every b-compact set is compact set in any topological space .
Proof :The proof is complete by remark ( 1.2 ) .
1.8 Definition [Maheshwari S.N. and Thakur S.S,1980]
A subset A of a topological space X is called α- compact set relative to X , if every
cover of A by α- open sets of X has a finite sub cover .
1.9 Theorem
Every b-compact set is α- compact set in any topological space .
Proof :The proof is complete by theorem (1.4).
1.10 Definition [W.Fleissner , J.Kulesza , and DR.Levy,1991]
A topological space ( X , T ) is said to be almost –compact if for each open cover
{W :   } ,there exist finite subsets i   such that X  {W :    i } .
1.11 Definition [Mustafa H.J. and Muhammad S.K. and Al-Kafaji N.T,2010]
A function f : X  Y is called αc-continuous if and only if for each x  X and
each open set V containing f (x) , such that V c is α-compact relative to Y , there
exists an open set U containing x such that f (U )  V .
1.12 Definition [P.E.Long and T.R.Hamlett ," H-continuous function,1975]
A function f : X  Y is called H-continuous if and only if for each x  X and
each open set V containing f (x) , such that V c is almost - compact relative to Y ,
there exists an open set U containing x such that f (U )  V .
1.13 Definition [Mustafa H.J. and Muhammad S.K. and Al-Kafaji N.T,2010]
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Journal of Babylon University/Pure and Applied Sciences/ No.(6)/ Vol.(21): 2013
The family of open sets having the complement α-compact relative to X may be
used as a base for a topology Tc .
2. bc-continuous function
2.1 Definition
A function f : X  Y is called bc-continuous if and only if for each x  X and each
open set V containing f (x) , such that V c is b-compact relative to Y , there exists an
open set U containing x such that f (U )  V .
2.2 Theorem
Every continuous function is bc-continuous .
Proof :Let f : X  Y be a continuous function , and let x  X , V be an open set containing
f (x ) such that V c is b-compact relative to Y .
Since x  X , V is an open set containing f (x) and f is continuous ,then there exists
open set U containing x such that f (U )  V .
Then f is bc-continuous function .
2.3 Theorem
For a function f : X  Y , the following are equivalent
1) f is bc-continuous .
2) if V is open in Y and V c is b-compact relative to Y , then f 1 (V ) is open in X .
3) if F is closed in Y and b-compact relative to Y , then f 1 ( F ) is closed in X .
Proof :1 2
Let V be an open set in Y such that V c is b-compact relative to Y , and let
x  f 1 (V )
Then f ( x)  V and there exists an open set U containing x such that f (U )  V
( from 1).
Then x  U  f 1 (V ) .
Then f 1 (U ) is an open set in X .
23
Let F be closed and b-compact set relative to Y .
c
Then F c is open and since F  F c is b-compact relative to Y .
Then f 1 ( F c ) is open in X ( from 2) .
Then ( f 1 ( F c )) c is closed in X , but ( f 1 ( F c )) c  ( f 1 (Y  F )) c
c
 ( f 1 ( F ))c  f 1 ( F )
Then f 1 ( F ) is closed set in X .
3 1
Let x  X and V be open set containing f (x) such that V c is b-compact relative to
Y.
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Since V is an open set , then V c is closed set in Y , and since V c is b-compact relative
to Y
Then f 1 (V c ) is closed set in X ( from 3) .
Then U  ( f 1 (V c )) c is open set containing x such that f (U )  V .
Then f is bc- continuous function .
2.4 Proposition
If A1 and A2 are b- compact sets relative to X , then A1  A2 is b-compact relative to
X.
Proof:Let F  W :   is an open cover of A1  A2 .
Then A1  A2 
W .
 
Then F is an open cover of A1 , A2 ,so for each i=1,2 there exists a finite subset  i of
 , such that Ai 
So, we have
W .
A  A  W
1
 i
2
, i=1,2
  i
Then A1  A2 is b-compact relative to X .
Let ( X , T ) be a topological space , it follows from proposition (2.4) that the family
of open sets having the complement b-compact relative to X may be used as a base
for a topology Tbc .
2.5 Remark
For a space X , we have Tc  Tbc  T .
2.6 Theorem
A function f : ( X , T )  (Y , T ) is bc-continuous if and only if f : ( X , T )  (Y , T bc ) is
continuous.
Proof :Suppose f : ( X , T )  (Y , T ) is bc-continuous function , and let A  Tbc
Then A is an open set in Y and Ac is b-compact relative to Y , so from theorem ( 2.3)
f 1 ( A) is an open set in X .
Then f is continuous .
Conversely
Let x  X and V be open set containing f (x) such that V c is b-compact relative to
Y.
Since f : ( X , T )  (Y , T  bc ) is continuous function .
Then f 1 (V ) is an open set in X containing x .
Let f 1 (V )  U ,then f (U )  V
Then f is bc-continuous .
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Journal of Babylon University/Pure and Applied Sciences/ No.(6)/ Vol.(21): 2013
The following theorems are immediate consequences of theorems (2.6) and the
proofs are omitted .
2.7 Theorem
If f : X  Y is bc-continuous and A is a subset of X , then the restriction
f / A : A  Y is bc-continuous .
2.8 Theorem
If f : X  Y continuous function and g : Y  Z is αc-continuous function ,then the
composition g  f : X  Z is bc-continuous function .
2.9 Theorem
If X is Hausdorff space and A is b-compact relative to X , then A is closed .
Proof :Let A be b-compact set relative to a Hausdorff space X .
So A is compact ( from theorem 1.7 )
Since X is Hausdorff , so A is closed set .
2.10 Theorem
Let Y is Hausdorff space ,then a function f : X  Y is bc-continuous if and only if
f 1 ( K ) is closed set in X for each set K is b-compact set relative to Y .
Proof :Let f : X  Y is bc-continuous , and let K is b-compact set relative to Y
Since Y is Hausdorff space , and f : X  Y is bc-continuous ,then K is closed set in
Y.
Then by theorem (2.9) f 1 ( K ) is closed in X .
Conversely
Since f 1 ( K ) is closed set in X for each set K is b-compact relative to Y
Then by theorem (2.3) f is bc-continuous .
2.11 Theorem
Let X is C1 - space , a subset A is b-compact relative to X if and only if A is almost
compact relative to X .
Proof:Let A is b-compact relative to X , and F  W :   be a cover of A , such that
W is an open set in X .
Then W is an b-open set in X from (1.2).
Since A is b-compact relative to X .
n
n
i 1
i 1
_
Then there exists 1 ,  2 ,...,  n in  such that A  Wi  W i
1998
n_
Then A  W i
i 1
So A is almost-compact relative to X .
Conversely
Let W :    be b-open cover of X .
_

Then W  :    is an open cover of X .


Since X is almost –compact , so there exists a finite subset  i   such that
_

X   W  :   i 


0
_
Since X is C1 - space ,then W   W is a finite for each    i .
Hence there is a finite subset F of X such that X  W :   i  F ,therefore X is
b-compact.
So A is b-compact set .
2.12 Theorem
Let Y be C1 - space , then a function f : X  Y is bc-continuous if and only if f is
H-continuous .
Proof :Let f : X  Y is bc-continuous, such that Y is C1 - space , and let x  X and V be
open set containing f (x) such that V c is almost –compact relative to Y .
Since Y is C1 - space, then V c is b-compact relative to Y , and since f is bccontinuous , Then there exists open set U containing x such that f (U )  V .
Then f is H-continuous .
Conversely
Suppose f is H-continuous , and let x  X and V be open set containing f (x) such
that V c is b-compact relative to Y .
Then V c is almost –compact from theorem (2.11) .
Since f is H-continuous ,then there exists open set U containing x such that
f (U )  V .
Then f is bc-continuous.
2.13 Theorem
Evry αc- continuous bc – continuous function is function .
The proof is complete from theorem (1.9).
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Journal of Babylon University/Pure and Applied Sciences/ No.(6)/ Vol.(21): 2013
References
Al Omari A. and M.S.Noorani ,2008, " Decomposition of Continuity Via b-open set "
Bol.Soc.Paran.Math.V.26(1-2),pp.53-64.
Andrijevic, D.1996, "On b-open sets ", Math.Vesnik,48(1996),pp.59-64.
Gaber,Sama Kadhim,2010,"On b-Dimension Theory",M.S.C,University of AlQadisiya .
Geentry K.R. and H.B.Hoyle.III ,1970," C-continuous function" , Yokohama. Math. J.
18 , pp.71-76 .
Maheshwari S.N. and Thakur S.S ,1980, " On α-irresolute Mappings ",Tamkang
J.Math.11,pp.209-214.
Mustafa H.J. and Muhammad S.K. and Al-Kafaji N.T. ,2010," αc-continuous function
" ,Journal of kufa for Mathematics and Computer, Vol.1,No.1,April,pp.104-111.
Njastad O.,1965," On Some Classes of Nearly Open Sets ",Pacific J.Math.15,pp.961970.
.Long P.E and T.R.Hamlett ,1975," H-continuous function ", Boll.Un.Math.Ital.(4),
11, pp.552-558.
Noiri T.,Al-Omari A. and M.S.M Noorani,"weak forms of W-open sets and
decomposition of continuity" ,EJp.A.M 2(1) (2009):73-84,.
Fleissner W., J.Kulesza , and DR.Levy , " Cofinality in normal almost compact space
", Proceedings of the American Mathematical Society ,Vol.113,No.2,1991.
2000