Rev. Acad. Canar. Cienc., XX (Núms. 1-2), 63-73 (2008) (publicado en septiembre de 2009)
q- C - SOMEWHAT CONTINUOUS FUNCTIONS
S. Ganguly & Bijoy Samanta
Department of Pure Mathematics, University of Calcutta
35, Ballygunge Circular Road, Kolkata-700019
India
e-mail: ganguly04@yahoo.co,in
Abstract :The aim of this paper is to introduce a new type of function called e - C
somewhat continuous function in a convex topological space is introduced earlier. Sorne
characterizations and various basic properties of this type of function are obtained. Also,
its relationship with other types of function is investigated. In this paper we have discussed
a comparison between a e - e somewhat continuous function and somewhat continuous
function.
AMS Subject Classification : 52A01, 54C10, 54E99.
Keywords:Convex topological space, T - e semi compatible, e - Csomewhat continuous
function, strongly C- regular space.
1 Introduction
The development of ·abstract convexity' has emanated from different sources in different
ways; the first type of development basically banked on generalization of particular problems
such as separation of convex sets [3], extremality [4]; [2], or continuous selection [10]. The
second type of development lay before the reader such axiomatizations, which, in every case of
design, express of particular point of view of convexity. With the view point of generalized
topology which enters into convexity via the closure or hull operator, Schmidt[1953] and
Hammer[1955], [1963], [1963b] introduced sorne axioms to explain abstract convexity. The
arising of convexity from algebraic operations and the related property of domainfitness
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received attentions in Birchoff and Frink[1948] Schmidt[1953] and Hammer[1963].
Throughout this paper the axiomatizations as proposed by M. L. J. Van De Ve! in his
papers in the seventies and finally incorporated in Theory of Convex Structure [12] will be
followed.
In [13] the author has discussed 'Topology and Convexity on the same set' and introduced
the compatibility of the topology with a convexity on the same underlying set. At the very
early stage of this paper we have set aside the concept of compatibility and started just with a
triplet(X,r, C)and have called it convex topological space only to bring back'compatibility'in
another way subsequently. With his compatibility, however, VanDevel has called the triplet
(X,r, C) a topological convex structure.
It is however seen that in many cases where compatibility is expected our definition serves
. the purpose.
In this paper, Art 2 deals with sorne early definitions and in Art 3, we have discussed
V-C space. Art.4 deals with B-Csomewhat continuous function and its basic properties. In
the last article a new type of convex topological space is introduced which is called strongly
C-regular space.
2 Prerequisites
Definition 2.1 [13] Let X be a nonempty set. A family C of subsets of the set X is called
a convcxity on X if
1.<I>,X EC
2. C is stable for intersection, i. e. if V ~ C is nonempty then nV E C.
3. C is stable for nested unions, i. e. if V ~ C is nonempty and totally ordered by set
inclusion then U'D E C.
The pair (X, C) is called a convex structure. The members of C are called convex sets
and their complements are called concave sets.
Definition 2.2 [13] Let C be a convexity on a set X. Let A~ X . The convex hull of A is
denoted by co(A) and defined by
co(A) = n{C: A~ CE C} .
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Note 2.3 [13] Let (X, C) be a convex structure and let Y be a subset of X. The family of
sets Cy = { C n Y : C E C} is a convexity on Y; it is called the relative convexity of Y .
Note 2.4 [13] The hull operator coy of a subspace (Y,Cy) satisfies the following: \fA ~
Y : coy(A) = co(A) n Y.
Definition 2.5 Let (X, T) be a topological space. Let e be a convexity 011 X. Then the
triplet (X,T,C) is called a convex topological space (CTS, in short).
Theorem 2.6 [5] Let (X, T , C) be a convex topological space. Let A be a subset of X.
Consider the set A., where A. is defined as follows : A. = { x E X : co(U) n A # </;, x E U E
T}. Then the collection T. = {A e : A ~ X, A = A.} is a topology on X su ch that T, ~ T.
A is said to be T.- closed if A = A •.
Definition 2. 7 [5] Let (X, T, C) be a convex topological space. The space (X, T , C) is called
T-C semi compatible if for every A ET, A. is a T.- closed set, i.e., if A ET, then (A.).= A •.
Definition 2.8 [5] Let (X, T , C1 ) and (Y, CJ, C2 ) be two convex topological spaces. A function
f: (X,T,Ci)-+ (Y,CJ,C2 ) is said to be
l. () - C-open if for each x E X and each nbd. U of x, there exists a nbd. V of f(x) in Y
such that V. ~ f(U.) and
2. () - C sornewhat open if U E T and U # </;, then there exists a V E CJ such that V # </;
and V. ~ f(U.).
3 V- C -space
Definition 3.1 A convex topological space (X, T, C) is said to be a V - C -space if every
nonernpty open subset of X is T. dense in Xi.e. if A(#</;) ET then A.= X.
Note 3.2 From the above definition it follows that if a subset A is T dense in Xi.e. A= X
then it is automatically T, dense in X.
Theorem 3.3 If a function f : (X , T, C1) -+ (Y, CJ, C2) is () - e somewhat open injection and
Y is a V - C -space, then X is also a V - C space.
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Proof: !et U be a any nonempty open set in X. Since f is (} - C somewhat open there exists
a nonempty open set V of Y such that V. <:;:; f (U.). Again since Y is a D - C -space, V. = Y.
So we have Y<:;:; f(U.). Now X= ¡-1(Y) <:;:; ¡ -1(f(U.)) =U. (since f is injective) and thus
we have U. = X. Consequently X is a D - C space.
Theorem 3.4 Let (X, T , Ci) be a D - e space and f : (X, T, C1) ---* (Y, a, C2) be a surjective
function. Then f is a B - C open iff it is B - C somewhat open function.
Proof: From the definition 2.9 of B-C open function and B-C somewhat open function it is
clear that if a function is B - C open then it is B - C somewhat open. To prove the converse
part, !et x E X and U be any nbd. of x. Then 3 O ET such that x E O<:;:; U.Since X is a
D -C space, O.= X and then we have U.= X.If we consider Y as a nbd. of f(x) , then we
get, Y. =Y= f(X) = f (U.). Hence f is(} - Copen function.
Definition 3.5 Let (X, T, C) be a convex topological space. A subset G of X is said to be
a T, - e closed if (Int(G)). = G.
Definition 3.6 A convex topological space (X, T , C) is said to be a (} - C irreducible space
if every pair of nonempty T, - C closed subsets of X has a nonempty intersection.
Remark 3. 7 Every D - C space X is a(} - C irreducible space.
Let H, G be two nonempty T, - C closed sets.Then H = (Int(H)) ., G = (Int(G)) .. since
Int(H), Int(G) are nonempty open sets and X is D-C space, (Int(H)). =X, (Int(G)). =
X. This shows that H íl G = X #- </>. Hence X is (} - C irreducible space.
The converse may not be true which follows from the next example.
Example 3.8 Consider the convex topological space (X, T, C) where X
{</>,X, {a}.{b,c}}, C ={</>,X , {a}}.
{a, b, c} , T =
Here (Int{a}). ={a}.= x, (Int{b}). = </>. = </>, (Int{c}). = </>. = </>,(Int{a,b}). ={a}.=
x, (Int{b,c}). = {b, c}. = {b,c} ,(Int{a,c}). ={a}.= x. So nonempty T, - C closed sets
are {b,c},X and {b,c}ílX = {b,c} #- <f>. Thus (X,T,C) is(} - C irreducible space. Now
{b, e} ET and {b, e}.= {b, e}#- X. Hence (X, T, C) is not D - C space.
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Proposition 3.9 Let (X, T, C) be a convex topological space which is T-C semi compatible.
For any subset A of X ' Int(A)). is T. - e closed.
Proof: Let V= (Int(A)) •. We will show that (Int(V)). =V.
Now Int(V) ~V which implies that (Int(V)). ~V.= ((Int(A).). = (Int(A)). (since X is
T - e semi compatible )=V.
Again !et x E (Int(A)). and x E U E T. Now Int(A) ~ (Int(A)). which implies that
Int(A) ~ Int((Int(A)) •. x E (Int(A)). =:? co(U)nlnt(A) .¡. </;=? co(U)nlnt((Int(A)).) .¡. </;
=:? x E [Int((Int(A)).)] .. Thus we have V~ (Int(V)) •. Hence V= (Int(V)) •.
Theorem 3.10 Let (X, T , C) be a convex topological space which is T - C semi compatible.
Then (X, T, C) is not a e - e irreducible space iff :3 nonempty open subsets u and V of X
such that u. n V. = </;.
Proof: Let X be not a e - e irreducible space. Then there exists nonempty T, - e closed sets
A and B of X such that A n B = </;. Since A and B are T, - C closed sets, (Int(A)). =A
and (Int(B)). =B. Let U= Int(A) and V= Int(B). Then U and V are nonempty open
sets such that U. n V. =</J. Conversely !et there exist a nonempty open sets U and V of X
such that U. n V. =</J. Let A= U. and B =V •. Then A= (Int(U)). and B = (Int(V)).
(by proposition3.9) are nonempty T, - e closed sets of X such that A n B = q;. Hence X is
not e - e irreducible space.
Theorem 3.11 Let f : (X, T, C1) -; (Y, O', C2) is e - e somewhat open injection where
(Y, O', C2 ) is O' - C semi compatible. If Y is e - C irreducible space then X is also a B - C
irreducible space.
Proof: Suppose that X is not a e - C irreducible space. Then there exists nonempty open
sets U and V of X such that U. n V. = </J. Since f is B - C somewhat open function there
exists nonempty open sets G and H of Y such that G. ~ J(U.) and H. ~ J(V.). Since
J is injective G. n H. = </J. This shows that Y is not a B - C irreducible space-which is a
contradiction. Hence X is a B - C irreducible space.
4 e - e somewhat continuous function
Definition 4.1 Lct (X, T, C1) and (Y, O', C2) be two convex topological spaces. A function
f: (X,T,C1)-+ Y,O", C2) is sáid to be
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l. () - C continuous function if for each x E X and each open nbd.V of f( x ), there exists an
open nbd. U of x such that f(U.) ~ V. and
2. () - C somewhat continuous function if V E a and ¡-1 (V) -/=- </>, there exists nonempty
open set U in X such that U. ~ ¡-1(V.).
Remark 4.2 From the above definition it follows that () - C continuity implies () - C somewhat
continuity. But the converse is not always true which follows from the next example.
Example 4.3 Let X= {a, b,c} , T = {</>,X, {a,b},{c}}, C1 = {</>, X, {a,b}, {c}}, a=
{</>,X, {a}, {e}, {a,c}, {b.c} }, C2 ={</>,X, {a}, {e}}. Here the identity function i: (X. r ,C1)--->
(X, a, C2 ) is B - C somewhat continuous function but not () - C continuous function. It is
clear that i is () - C somewhat continuous function. Consider the point b E X. Now { b, e} is
nbd. of b = i(b) in (X,a,C2 ) and in (X,a,C2 ), {b,c}. = {b,c}. Again nbd. of b in (X,r,C1)
are X and {a,b}. In this space X.= X and {a,b}. = {a,b}. But {a,b} <l i- 1{b,c} = {b,c}
and X <l i-1 {b, e} = {b, e}. Consequently i is not B - C continuous function.
Theorem 4.4 Composition of two B - C somewhat continuous functions is again B - C
somewhat continuous function.
Proof: Obvious.
Theorem 4.5 Let f : (X, r, Ci) ---> (Y, a, C2 ) be a() - C somewhat continuous surjection. If
(X, T, C1) be a D - e space then (Y, a, C2) is also a D - e space.
Proof: Let V(# </>) E a. Since f is surjective , ¡-1(V) -/=- <f>. As f is () - C somewhat
continuous function, there exists U(# </>) E T such that U. ~ ¡ -1(V. ). Now X is a D - C
space. So U. =X. Hence Y= f(X) = f(U.) ~ ff- 1(V.) =V. i.e. we have V. = Y. This
shows that Y is a D - C space.
Theorem 4.6 Let f: (X,r,C1 )---> (Y,a,C2 ) be a B-C somewhat continuous function where
(X, T , C1) is 'T - e semi compatible. If X is a() - e irreducible space then y is also a() - e
irreducible space.
Proof: Let Y be not () - C irreducible space. Then there exist nonempty open sets U and V
in Y such that U. n V. = <j;. Since f is () - C somewhat continuous, there exist nonempty
open sets G and Hin X such that G. ~ ¡-1(U.) and H.~ ¡-1(V.). This implies that
G. n H. ~ ¡-1(U.) n ¡ - 1(V.) = ¡-1(U. n V.) = <j;. This shows that X is not a () - C
irreducible space-which is a contradiction. Hence Y is a() - C irreducible space.
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Result 4.7 [5] Let (X, T, C) be a convex topological spacc and A<:;; X. Considcr the convex
topological space (A, TA' c A) where TA is subspace topology and c A is relative convexity on
A. Then for any subset B of A , ((B).)'"" <:;; B •.
Theorem 4.8 Let f: (X,T,C1)----> (Y,cr,C2) be a {}-C somewhat continuous function and A
be a dense subset of X. Then f: (A, TA,CA)----> (Y, cr,C2) is also {}- C somewhat continuous
function.
Proof: Let V E cr such that ¡-1(V) =/= </;. Since f: (X, T,C1)----> (Y, cr,C2) is a {}-C somcwhat
continuous function, there exists U(=/=</;) E T such that U. <:;; ¡-1(V.). A is dense in X
implies that AnU =/= </;. Now Un A E TA. By result 4.7 we have (UnA);A <:;;(Un A).-<:: U •.
So (Un A):" <:;; u. <:;; ¡-1(V.). This shows that f : (A, TA , CA) ----; (Y, cr, C2) is also () - c
somewhat continuous function.
Theorem 4.9 Let f : (X, T , C1) ----> (Y, cr, C2) be a function where (Y, cr, C2) is 1J - C-space.
Then J is () - C continuous function iff () - C somewhat continuous function.
Proof: For any function f : (X, T, C1) ----; (Y, cr, C2) it is clcar that () - c continuity => () - c
somewhat continuity [by Remark 4.2].
Conversely let J be () - C somewhat continuous function and Y is a 1J - C-space.
Let x E X and V be open nbd. of J(x) in (Y, cr,C2 ). since Y is a 1J - C-space, V. = Y.
In (X, T, C1 ) , we take X as a open nbd. of x. Then clearly J(X.) = f(X) <:;; Y = V..
Consequently J is () - C continuous function.
Result 4.10 Let (X, T, C) be a convex topological space. Let A E T. Then (A.)7" <:;; (A.)7"".
Proof: Consider the convex topological space (X,TA ,CA)· We have to prove that (A.)'"<:;;
(A.)'"A. Let x E A. and let V E TA such that x E V E TA. Since V E TA, V= AnU for some
U E T. This shows that V E T. Now x E (A.)7" => co(V) n A=/=</; => co(V) n A n A=/=</; =>
coA(V) n A=/= </; [by relative hull formula] => x E (A.)7"A. Hence (A.)7" <:;; (A.)'"A.
Theorem 4.11 Let (X, T, C) and (Y, cr, C1) be two convex topological spaces. Let A be an
open subset of X such that f : (A, TA, CA) ----> (Y, cr, C1) is {}-C somewhat continuous function
and f(A) is dense in Y. Then any extension F off is() - C somewhat continuous function.
Proof: Let U be any open set in Y such that F-1(U) =/= </;. Since J(A) is dense in Y,
Un f(A) =/= </; and F - 1(U) n A =/= </;. That is ¡-1(U) n A =/= </;. Since f : (A, TA, CA) ---->
(Y, cr, C1) is () - C somewhat continuous function and U E cr with ¡ -1 =/= </;, there exists
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V E TA with V -=J <P such that (V.YA ('.;;; ¡ - 1(U.)- - (l). It is clear that V is open in X
as A E T. Thus by result 4.10 we have (V.Y ('.;;; (V.Y"· Consequently from (1) we have
(V.y ('.;;; (V.YA ('.;;; ¡-1(U.) ('.;;; F - 1 (U.). This shows that F is e - C somewhat continuous.
Theorem 4.12 Let (X, T, C1) and (Y, O", C2) be two convex topological spaces. Suppose
X= A U B where A andB are open subsets of X and f : (X, T, C1 ) -> (Y, O", C2) is a function
such that f IA and f Is are e - e somewhat eontinuous function. Then f is e - e somewhat
continuous function.
Proof: Let U E O" such that ¡ -1(U) -=J ¡p. Then (f 1At1(U) -=J <Por (f l8 )-1(U) -=J <P or both
(f IA)-1(U) -=J <P and (f ls)- t(U) -=J ¡p.
Case l. (f IA)-1(U) -=J ¡p. Since f IA: (A, TA, CA)-> (Y,O",C2) is e-e somewhat continuous,
there exists V E TA with V -=J <P such that (V.YA ('.;;; (f IA) - 1(U.) ('.;;; ¡- 1(U.). As V ET by
result 4.10 (V.y('.;;; (V.YA('.;;; ¡-1(U.). This implies that f: (X,T,C) _, (Y, O",C2) is e- e
somewhat continuous function. Similarly for the other case.
Definition 4.13 Let (X, C) be a convex structure and let T1 and T2 be two topology on X.
Then Ti is said to be e - e weakly equivalent to T2 provided if u E T] and u -=J <P, thcn therc
exists a nonempty set V E T2 such that (V.)7" ('.;;; (U.Y' and if PE T2 and P -=J <P, then there
exists a nonempty set Q E T1 such that ( Q. y1 ('.;;; (P. y2 •
Note 4.14 Consider the identity function i: (X,T1,C)-> (X,T2 ,C) and let T¡ and T2 be
weakly equivalent. Let V E T2 such that i - 1(V) = V -=J ¡p. Then there exists U E T 1 with
U -=J <P such that (U.y1 ('.;;; (V.y2 => (U.y1 ('.;;; (V.)72 = i-t((V.)1"2). This shows that i is e - C
somewhat continuous function. Similarly we can show that i: (X, T2,C) -> (X,T1,C) is also
e - e somewhat continuous function.
Conversely if i : (X, T1, C) -> (X, T2, C) is e - e somewhat continuous function in both
directions then T1 and T2 are e - C weakJy equivalent.
Theorem 4.15 Let f : (X, T1, C) -> (Y, O", C) be e - e somewhat continuous function. Let
T2 be a topology 011 X which is e - e weakly equivalent to T1. Then f : (X, T2 , C) -> (y, O", C)
is e - e somewhat continuous function.
Proof: Obvious.
Theorem 4.16 Let f : (X, T, C) -> (y, O", C) be e - e somewhat continuous function. Let (/¡
be a topology on y which is e - e weakly equivalent to(/. Then f : (X, T , C) -> (y, O"i, C) is
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also () - e continuous function.
Proof: Obvious.
5 Comparison between 8-C somewhat continuous function
and somewhat continuous function
In this articlc wc will show that a fJ - C somewhat continuous function is not ncccssarily a
somewhat continuous function and vice versa. Here we also show that there exists a special
type of convex topological space in which the above two concept coincides.
Example 5.1 Let X= {a, b, c, d}, T = {</>,X, {b}, {d}, {b, d}}, T1 = {</>,X, {a, b}, {e}, {a, b, e}},
C = C1 = {<f;,X} and consider the function f: (X,T,C)......, (X,T1,C1) defined by f(a) =e,
f(b) = a f(c) = e f(d) = b. Here f is fJ - C sorne what continuous function but not
somewhat continuous function. Note that for any U E T1 , in the convex topological space
(X,T1,C1), U.= X. Now {e} E T1 and ¡ -1({c}) = {a,c} =J </>, but there is no V ET such
that V~ ¡-1({c}).
Example 5.2 Let X= {a, b, e}, T = {</>,X, {a} , {b}, {a, b}}, C = {</>,X}, C1 = {</>,X, {a}}
and consider the idcntity function f : (X,T,C)......, (X,T1,C1). It is clear that f is somcwhat
continuous function. Now {b} E T1 and in thc convex topological space (X, T1 , C1 ), {b }. =
{a, e}. Again in the convex topological space (X, T, C) for any U E T, U. = X. Thus for
{ b} E T1 , thcre is no V E T su ch that v. ~ ¡-1 ( { b}.). Conscqucntly f is not fJ - C somcw hat
continuous function.
Definition 5.3 A convex topological spacc (X, T, C) is said to be strongly e-regular space
if for any nonempty set U ET there exists a nonempty set V E T such that V. ~ U.
Example 5.4 Let us consider the convex topological space (X, T, C), where (X, T) is discrete
topological space and C is defined by C = {<f;,X} U {{x}: x E X}. then for any U ET,
U.= U and thus (X, T,C),is strongly C-rcgular space.
Example 5.5 Let R denote the set of reals and Tu be the usual topology on R. Here the
convexity C is defined as follows : A set C E C iff for any two points a, b E C, the convex
combination of a, b must be in C i.e. C is the standard convexity on R. Then it is clear that
( R, Tu, C) is a strongly C-regular space.
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Example 5.6 Any locally convex space (X, í) is strongly e-regular space. Already we have
proved in [5] that in a locally convex space, if A = A then A = A = A.. Let V be any
nonempty open set in X. Since (X, í) is regular space, there exists W E T with W # </J,
such that W ~V. So we have (W). = W ~V. This shows that W. ~ (W). ~V. So we see
that for any nonempty open set V , there exists a nonempty open set W such that w. ~ V.
Consequently (X, í) is strongly e-regular space.
Theorem 5. 7 Let f : (X, T , e 1) -+ (Y, O', e2 ) be a e-e somewhat continuous function which
is onto. If (Y, O', e 2 ) be strongly e -regular spacc, then f is a somewhat continuous function.
Proof: Let U E O' with U # </J. Then ¡-1 (U) # </J. Now U E T and (Y, O', e2 ) is strongly
e-regular space so there exists W E O' with W # </J such that lV. ~ U. Since f is e - e
somewhat continuous function and ¡-1(W) # </J, there exists V E T with V # <P such that
V. ~ f - 1(W.). This implies that V~ v. ~ ¡ -1(lV.) ~ f -1 (U). Consequcntly f is somewhat
continuous function.
Theorem 5.8 f : (X, T, C) -+ (Y, O', e 1) be a somewhat continuous function. If (X, T, e ) be
a strongly e-regular space, then J is e - e somcwhat continuous function.
Proof: Let U E O' and ¡-1(U) # </J. Sincc f is somcwhat continuous function thcrc cxists
V ET with V # <P such that V~ ¡-1(U). Again (X,T,e) is a strongly e-regular space
so for V E T there cxists W E T with W # <P such that W. ~ V. This shows that
w. ~ V~ ¡-1(U) ~ ¡-1(U.). Therefore J is e - e somewhat continuous function.
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