Some functions related to convextopological spaces

              Throughout this paper the axiornatizations as proposed by M. L. J. Van De Vel in his
papers in the seventies and finally incorporated in Theory of Convex Structure [11] will be
followed.
In [12] the author has discussed 'Topology and Convexity on the same set' and introduced
the cornpatibility of the topology with a convexity on the sarne underlying set. At the very
early stage of this paper we have set aside the concept of cornpatibility and started just with
a triplet(X,T, C)and called it convex topological space only to bring back'cornpatibility'in
another way subsequently. With his cornpatibility, however, VanDevel has called the triplet
(X, T, C)a topological convex structure.
It is however seen that in rnany cases where cornpatibility is expected our definition serves
the purpose.
In this paper, Art 2 deals with sorne early definitions and Art 3 envisages a new topology
generated on a convex topological spaces via convexity; by (X,T, C)is a convex topological
space ( CTS in short) then the generated topology T, , in general, is such that T, t;;; T; it has
been shown by an exarnple that T, e T. In Art 4, sorne new types of functions have been
introduced and in Art 5, we have introduced a special type of convex topological space. The
last Art 6 deals with sorne basic properties of () - C sornewhat open function.
2 Prerequisites
Definition 2.1 [12] Let X be a nonernpty set. A family C of subsets of the set X is called
a convexity on X if
1.<l>,XEC
2. C is stable for intersection, i. e. if D t;;; C is nonernpty then nD E C.
3. C is stable for nested unions, i. e. if D t;;; C is nonernpty and totally ordered by set
inclusion then UD E C.
The pair (X, C) is called a convex structure. The rnernbers of C are called convex sets
and their cornplernents are called concave sets.
Definition 2.2 [12] Let C be a convexity on a set X. Let A t;;; X. The convex hull of A is
denoted by co( A) and defined by
co(A) = n{C: A t;;; CE C}.
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Note 2.3 [12] Let (X, C) be a convex structure and let Y be a subset of X. The family of
sets Cy = {C n Y: CE C} is a convexity on Y; it is called the relative convexity of Y.
Note 2.4 [12] The hull operator coy of a subspace (Y, Cy) satisfies the following : VA ~
Y : coy( A) = co(A) n Y.
Definition 2.5 Let (X, T) be a topological space. Let e be a convexity on X. Then the
triplet (X,T,C) is called a convex topological space (CTS, in short).
3 N ew Topology T* on CTS
In this article we have introduced a new topology T. on CTS with sorne examples on such
topologies.
Let (X, T, C)be a CTS. Let A~ X. We consider the set A., where A. is defined as follows
: A.={xEX:co(U)nAyf<l>,xEUET}.
From the definition of A., it is clear that A ~ A ~ A •.
Proposition 3.1 Let (X, T , C) be a CTS; then
l. <I>. = <I>, X. =X.
2. A, B ~ XandA ~ B thenA. ~B •.
3. (A U B). =A. U B •.
Proof:l. Obvious.
2. Obvious.
3. A~ (AUB) =}A.~ (AUB). and B ~ (AuB) =}B.~ (AUB) •. So A.uB. e (AUB) •.
To prove the converse part, !et x E (A U B) •. If possible let x rf. A. and also x rf. B •. Then :3
U1 and U2 such that co(U1) n A= <I> and co(U2) n B = <I> where x E U1 ET and x E U2 E T.
Now x E U1 n U2 ET and co(U1 n U2) ~ co(U1) also co(U1 n U2) ~ co(U2).
Hence co(U1 n U2) n (A U B) = (co(U1 n U2) n A) U (co(U1 n U2) n B) = <I>-which contradicts
the fact that x E (A U B) •. Hence either x E A. or x E B., i. e. (A U B). ~ A. U B •.
Consequently (A U B). = A. U B •.
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Theorem 3.2 Let us consider the collection T, = {A° : A~ XandA =A.}. Then T, is a
topology on X such that T, ~T.
Proof: l. ef;. = if; and X.= X , so if;, X ET,.
n
2. Let A1, A2, - - -An ET, and B = ílAi·
i=l
n
Now (Be).= [( íl A;)"].
i=l
n n
= [LJAf]. = LJ (Af).[by Proposition 3.1]
i=l i=l
n
= LJAf [since Ai E T., (AT). = Af]
i=l
= (íl Ai)e =Be.
Thus B = (Be)e E T,.
3. Let A,,(a E A) ET,. Let B = LJ A,,.
aEA
First we prove that ( íl A~). = íl A~.
a EA aEA
Now íl A~~ ( íl A~) •.
aEA a EA
Again íl A~~ A~; Va E A.
a EA
=> ( íl A~). ~ (A~). = A~ [since A,, E T,] Va E A.
a EA
=> ( n A~). ~ n A~.
a EA a EA
Hence ( íl A~).= íl A~. Now, (Be).= (( LJ Aa)°). = íl A~= íl A~ = ( LJ A,,)e = Be
a EA aEA a EA aEA a EA a EA
and so B = (Be)e ET,. Thus T, is a topology on X .
Let A ET, and Jet B =A°. Then we have B = B •. Since for any B ~X, B ~ B ~B.,
we haveB = B = B •. Thus Be= A E T. Hence T, is a topology on X such that T, ~T.
Note 3.3 The members of T, are called convex-open sets and a set A~ Xis called convex­closed
if Ae E T •.
In the following examples we will show that T. may be strictly coarser than T .
Example 3.4 Let X = [-1, l]. Let a topology Ton X consists of those subsets of X which
either do not contain {O} or contain (-1, 1).
Let a convexity C on X be defined as follows:
e = { x, <I>} u {{ x} : x E x}.
It is clear that, {1} , { - 1 }, { - 1, 1} and any set containing {O} are T closed sets. Now O E {1}.,
O E {-1}. and O E {-1, 1}. so that {1}. -=f. {1},
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{-1}. # {-1},
{-1, 1}. # {-1, 1 }. Again !et A be any subset such that O E A. Let p rf. A. Then
{p} #{O}. Since {p} ET as well as {p} E C, we claim that p rf. A •. Hence A.= A. Hence
the new topology T• consists of ali those subsets which do not contain {O} together with X.
Thus {<I>,X} e T. e T.
Example 3.5 Let X = {a, b, e}.
T = { <I>, X , {a}, { b}, {a, b}, { b, e} }.
C = {<I>,X,{a},{b}}.
Here, T.= {<I> ,X , {b}, {a}, {a, b}}. Hence, T. e T.
In CTS, we see that the topology T. is coarser than the original topology and the above
examples show that T. may be strictly coarser than T. So it is natural to ask whether, for
an arbitrary set X , there exists a topology anda convexity on X for which T. coincides with
the original topology. The following examples serve the purpose.
Example 3.6 Let X be any set and T = P(X). Let C be consists of <I>, X and ali singletons.
Here for any set A<:;; X, we have A= A •. Hence in this case, we have T.= T = P(X).
Again for any set X , if T = {<I>, X} and C be any convexity on X, then T.= {X, <I>} =T.
These are the trivial examples for which the topology T. coincide with the original topology.
We now give another non trivial interesting example.
Example 3. 7 Let X be a set and C be any convexity on X. From the definition of convexity
e on X, it is clear that e is a base for sorne topology. Let this topology be denoted by T(C).
Now !et us consider the convex topological space (X, T(C), C). Since T. <:;; T(C), we only
show that every T(C) closed sets are also T. closed. Let A be T(C) closed, i.e. A =A. To
prove A = A., it is sufficient to show that A. <:;; A[since A = A <:;; A.]. Let x E A. and
x E U E T(C). If U E C, then co(U) n A= Un A# </>[since x E A.]. Again if U E T(C) \ C,
then 3C E e such that X E e<:;; u. Now CnA = co(C)nA # </>[since X E A.]=> Un A#</>.
Thus x E A and we get A= A= A •. Consequently T.= T(C).
4 e - C Somewhat Open Functions
In this article we define new types of functions on CTS with examples. Also we discuss their
mutual relationship.
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Definition 4.1 [5] Let (X, O") and (Y, T) be topological spaces. A function f : (X, O") --+
(Y, T) is said to somewhat open provided if U(#- <I>) E O", then 3 a V ET such thatV #- <I>
and V t:::; J(U).
Definition 4.2 A function f : (X, T , C1) ---+ (Y, v, C2 ) is said to be B-C-open if for each
x E X and each nbd. U of x, 3 a nbd.V of f(x) in Y such that V. t:::; f(U.).
Definition 4.3 A function f : (X, T, C1) ---+ (Y, v, C2) is said to be B-C-somewhat open
[respectively almost somewhat C-open, weak somewhat C-open] (briefly B-C. sw. o, a. sw.
C. o, w. sw. C. o respectively) if U E T and U#- <I> then 3 a V E v such that V #- <I> and
V. t:::; f(U.) [respectively V t:::; f(int(U.)) , V t:::; J(U.)]
Remark 4.4 We obtain the following diagram from the definitions.
DIAGRAM- I
open f~ ==> somewhat open f~ ==> almost somewhat C-open f~
B-C-open f~ ==> B-C-somewhat open f~ ==> weak somewhat C-open f~
Remark 4.5 The following examples enable us to realize that none of these implications is
reversible.
Example 4.6 Let X= {a,b,c}, T = {<I> ,X, {a} , {c} , {a,c}, {b,c}} , C1 = {<I>,X,{a},{c}},
v = { <I> , X , {a, b} , {e}}, C2 = {<I> , X , {a, b} , {e}} and the function f : (X, T,C1) --+ (X, v, C2)
be defined as follows : J(a) = a, J(b) = b, J(c) = c. Here f is e - C-somewhat open
function. Now consider the point b in (X,T,C1). {b,c} is a nbd.of b in (X, T, C1) . But in
(X,v,C2), X.= X and {a,b}. = {a ,b} and {a ,b}. 7= J({b,c}) , X. 7= f({b,c}). Hence f is
not e - C-open function.
Example 4.7 Let X= {a, b, e}, T = {<I>, X, {b} , {e}, {b, e}}, C1 = {<I> , X, {a}}, v = {<I>, X, {e}, {
C2 = { <I>, X , { c}}and the function f : (X, T, C1) --+ (X, v, C2) be defined as follows : J(a) =e,
f(b) =a, f(c) = b.
Here J is e - C-open, B - C-somewhat open function , but J is not open and not somewhat
open function.
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For the convex topological space (X, r, C1), {b}. =X , {e}. =X, {b, e}. =X. Thus for any
U E r(=I- <P) , we take V= X in (X,v,C2 ) such that V.~ f(U.). So f is(} - C-somewhat
open function.
Since for any U E r(=I- <P), U,= X, we can say that f is(} - C-open function.
Here {e} E r but f({c}) = {b} .;_v. So f is not open function. For {b} E r, since 3 a
nonempty V E v such that V Ck_ f ( {b}) = {a}, f is not somewhat open function.
Example 4.8 Let X= {a,b}, T = {<P,X, {a} , {b}}, C1 = {<P,X, {a}}, Y= {x ,y,z},
v = {<P,Y, {x},{y},{x,y}}, C2 = {<P,Y}, and the function f: (X,r,C1)----> (Y,v,C2 ) be
defined as follows : f(a) = x, f(b) =y. Here f is open but not (} - Copen function. Clearly
f is an open function. In the convex topological space (X,r,C1), {b}. = {b}.
In the convex topological space (Y,v,C2), {x}. =Y, {y}.= Y, {x,y}. =Y.
So if we take U = {b} E T then there is no V E v such that V. ~ f (U.).
Hence f is not (} - C-somewhat open function. Consequently f is not (} - C-open function.
Example 4.9 Let X= {a, b}, r = { <P, X , {a}}, C1 = {<P, X}, v = {<P, X}, C2 = {<P, X}and
the function f: (X, r, C1) ----> (Y, v, C2) be defined as follows : f(a) = b, f(b) =a.
Here f is almost somewhat C-open but not somewhat open function.
In the convex topological space (X,r,C1 ), X. = X and int(X.) = int(X) = X , {a}.= X
and int( {a}.) = int(X) = X . Hence clearly f is almost somewhat C-open function (take
V =X).
Here {a} E r but X Ck_ {b}, i.e., X Ck_ f({a}). Hence f is not somewhat open function.
Example 4.10 Let X= {a,b,c}, T = {<P ,X ,{a} , {b} ,{a, b}}, C1 = {<P,X,{a}}, v =
{ <P, X, {e}}, C2 = { <P, X} and the function f : (X, r, C1) ----> (Y, v, C2) be defined as follows:f (a) =
a, f(b) = b, f(c) =c.
Here f is weak somewhat C-open but not (} - C somewhat open and not almost somewhat
C-open function.
In the convex topological space (X,r,C1 ), {a}.= X, {b}. = {b,c}, {a,b}. =X. If U= {b}
then take V= {e} E v such that V ~ f ( {b}.).
Again if U = {a} or, U = {a, b} , then take V = X. So f is a weak somewhat C-open
function.
Now in the convex topological space (X, T, C2), {e}. =X and X. = X. Thus for U= {b} ET,
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there is no V E v such that V.~ J(U.)[J(U.) = J({b.}) = J({b ,c}) = {b,c}].
Hence f is not () - C somewhat open function.
Again for U= {b} ET, {b}. = {b, c} and int(U.) = int({b}.) = int({b,c} = {b}. But there
is no V E v such that V~ f(int(U.)).
Hence f is not almost somewhat C-open function .
Example 4.11 Let X= {a ,b} , T = {<P ,X, {a}} , C1 = {<P ,X}, v = {<P ,X}, C2 = {<P ,X}
and the function f: (X, T, C1) -> (X, v, C2) be defined as follows : J(a) =a, J(b) = b.
Here f is almost somewhat C-open function but not somewhat open function.
In the convex topological space (X,T,C1), {a}.= X. So, int({a}.) = int(X) =X. Thus for
U = {a} ET, take V= X E v such that V~ f(int(U.)). Hence f is almost somewhat C
open function.
Again for U= {a} ET, there is no V E v such that V~ J(U). Hence f is not somewhat
open function.
Example 4.12 LetX = {a ,b,c},T = {<P ,X, {a} , {b,c}} ,C1 = {<P ,X , {a}} ,v = {<P ,X, {a},{b, c}},
C2 = {<P ,X} and the function f: (X,T,C1)-> (X,v,C2) be defined as follows: J(a) =a,
f(b) = b, f(c) =c.
Here f is somewhat open function but not () - C somewhat open function. Clearly f is open
function and hence f is somewhat open function.
In the convex topological space (X, v,C2) , {a}.= X , {b,c}. =X. Thus for U= {b,c} ET,
we have no V E v such that V.~ f(U.) [since {b,c}. = {b,c} in (X,T,C1)]. Hence f is not
() - e somewhat open function.
5 Special type of Convex Topological Space
In this article we define a spacial type of convex topological space. Also we introduce new
types of CTS on which reverse implications of the diagram-1, as shown in the previous article
holds.
Definition 5.1 Let (X, T , C) be a convex topological space. The space (X, r, 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,.
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Example 5.2 Let X= {a ,b,c}, T = {<l>,X, {a}, {b}, {a, b}} , C = {<l>,X, {a}}. Here {a}.=
X , {b}. = {b,c} and {b,c}. = {b,c}. Thus [{b}.]. = {b} •. Again {a , b}. =X. Thus
(X, T, C) is T - C semi compatible.
Example 5.3 Let X= [-1, l]. Let T consists of ali those subsets of X , which either do not
contain {O} or contain (-1, 1). Let C = {<l>,X} U {{x}: x E X}. Here (-1, 1). = (-1, 1),
(-1, 1]. = (-1, 1], [-1, 1). = [-1, 1). Let A ET be such that O rf. A. Now A.= A U {O}
and [A U {O}]. =A U {O}. So for every A E T , A. is T,-closed. Thus (X, T, C) is T - C semi
compatible.
Example 5.4 Every normed linear space with usual convexity is T - C semi compatible. In
fact, every locally convex space is T - C semi compatible.
Let (X, T) be a locally convex space. Let us consider the topology T, on (X, T). Again !et
A~ X be such that A= A. We will show that A= A,. Let x E A. and x E U E T. Since
(X, T) is locally convex space, there exists V E T such that V is convex and x E V ~ U.
Now x E A. :::;. co(V) n A =!= <P :::;. V n A =!= rjJ[since V = co(V)] :::;. Un A =!= <P :::;. x E A.
Consequently A = A = A.. Thus in this case we have T = T,. Hence (X' T) is T - e semi
compatible.
In the following example we will show that not ali convex topological space (X, T, C) is
T - C semi Compatible.
Example 5.5 Let X= {a ,b,c, d,e}, T = {<1>,X,{d} ,{e},{e, d}} , C = {<l>,X,{c,d}}. Here
{e} E T. Now {e}. = {a, b, e, e} and ( {e}.). =X=!= {e} •. Hence (X, T, C) is not T - C semi
compatible.
Definition 5.6 A subset G of a CTS (X, T, C) is said to be R - C open if int( G,) = G.
Proposition 5. 7 l)In (X, T , C) every R - e open set is open.
2)If (X, T, C) is T - C semi compatible, then for every A ET, int(A.) is R - Copen.
Proof:l) Obvious.
2) Let A ET and !et V= int(A.). Since (X, T , C) is T - C semi compatible, (A.). =A. We
will show that int(V.) = V. Now int(V.) = int((int(A.)).) ~ int((A.).) = int(A.) =V.
Again V= int(V) ~ int(V.). Hence int(V.) =V.
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Definition 5.8 A CTS (X, T, C) is said to be a
1) QR - C space if for every nonempty open set G of X , there exists a nonempty open set
D such that D. <;;; G.
2) SQR - C space if for every nonempty open set G of X , there exists a nonempty R - C
open set A such that A<;;; G.
3) AQR - C space if for every nonempty R - Copen set A<;;; X, there exists a nonempty
open set D such that D. <;;; A.
Proposition 5.9 A CTS (X, T, C) which is T - e semi compatible is a QR - e space iff it
is both SQR - C space andan AQR- C space.
Proof : Let (X, T, C) is a Q R - C space. Again let G be a nonempty open set of X. Since X
is QR- C space, there exists a nonempty open set D such that D. <;;; G. Thus int(D.) <;;; G.
Since int(D.) is R - C open set [by Proposition 5.7.(ii)] we conclude that X is SQR - C
space.
Again let A be any nonempty R - Copen set. Then A is an open set. Since X is a QR- C
space, there exists a nonempty open set D of X such that D. <;;;A. Hence X is a AQR - C
space.
To prove the converse, let G be any nonempty open set of X . Since X is SQ R - C space,
there exists a nonempty R - C open set A of X such that A <;;; G. Again since X is a
AQR - C space, there exists a nonempty open set D of X such that D. e A. Hence
D.<;;; A<;;; G.Consequently X is a QR - C space.
Note 5.10 For the proof of the converse part of the above Proposition the requirement, it
is evident that T - C semicompatibility of (X, T, C) is not necessary.
Theorem 5.11 Let f : (X, T , C1) ---+ (Y, v, C2) be a function. Then the following properties
holds:
1) If (Y, v, C2) is QR - C space, then f is () - C somewhat open iff it is weak somewhat-C
open.
2) If (X, T , C1) is T - e semi compatible and AQR - e space, then f is almost somewhat-C
open iff it is weak somewhat-C open.
3) If (X, T, C1) is SQR-C space, then f is somewhat open iff it is almost somewhat-C open.
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Proof : 1) If f is e - C somewhat open then it is clearly [follows from the diagram] weak
somewhat-C open. To prove the converse, let U be any nonempty open set of X. Since
f is weak somewhat-C open function, there exists a nonempty open set V of Y such that
V ~ f(U.). Again since Y is QR - C space, there exists a nonempty open set D of Y such
that D.~ V. Thus we have D.~ V~ f(U.). Consequently f is a e - C somewhat open.
2) If f is almost somewhat-C open then clearly it is weak somewhat-C open. Let A be any
nonempty open set of X. Since A(# cp) ET and A~ A., we have int(A.) # cp. Now int(A.)
is R - C open set. Since X is AQ R - C space, there exists a nonempty open set D of X such
that D. ~ int(A.). Again since f is weak somewhat-C open, there exists a nonempty open
set W of Y such that W ~ f(D.) . Thus we obtain that W ~ f(D.) ~ f(int(A.)). Hence f
is almost somewhat-C open function.
3) If f is somewhat open then clearly it is almost somewhat-C open. To prove the converse,
let X be an SQR - C space. Again let G be any nonempty open set in X. Since X is
SQR-C space, there exists a nonempty R-C open set A of X such that A~ G. Now A is
an open set and since f is almost somewhat-C open, there exists a nonempty open set W of
Y such that W ~ f(int(A.)) = f(A) [since A is R - Copen, int(A.) =A]. Hence we obtain
that W ~ f(A) ~ f(G). Therefore f is a somewhat open function.
Corollary 5.12 If (X,T,C1) is T-C semi compatible and QR-C space, then the following
concepts on a function f : (X, T , C1) ----> (Y, v, C2) : somewhat open, almost somewhat open
and weak somewhat C open are equivalent.
Corollary 5.13 If (X,T,C1) and (Y,u,C2) are QR- e space and (X,T,C) is T - e semi
compatible, then the following concepts on a function f : (X, T , C1) ----> (Y, V, C2) : e - e
somewhat open, somewhat open, almost somewhat C open and weak somewhat C open are
equivalent.
6 Properties of e - C somewhat open function
Theorem 6.1 If f: (X, T , C1) ----> (Y, v, C2) and g: (Y, v, C2) ----> (Z, 7/J, C3 ) are B-C somewhat
open functions, then g o f : (X, T, C1) ----> (Z, 7/J, Ca) is also e - e somewhat open function.
Proof: Let U be any nonempty open set of X. Then there exists a nonempty V E v such that
V. ~ f (U.) because f is e - C somewhat open. Again since gis e - C somewhat open, there
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exists a nonempty W E 'l/; such that W. ~ g(V.). Hence W. ~ g(V.) ~ g(J(U.)) = (gof)(U.).
This shows that g o f is () - C somewhat open.
Result 6.2 Let (X, T , C) be a convex topological space and A ~ X. Consider the convex
topological space (A, TA, CA) where TA is subspace topology and CA is relative convexity on
A. For any subset B of A, (B);A ~ B •.
Proof : Let X E (B):A and X E u E T . It is clear that X E A, so X E u n A. Now
x E Un A E TA=> coA(U n A) n B f cp => co(U n A) n A n B f cp [by relative hull formula]
=> co(U n A) n B f cp [since B ~ A] => co(U) n B f cp [since Un A~ U, co(U n A) ~ co(U)]
=> x E B •. Hence (B);A ~B •.
Theorem 6.3 If (X, T, C1) and (Y, v, C2) are convex topological spaces and A is dense in X
and f : (A, TA, C1A) ----> (Y, v, C2) is () - C somewhat open then any extension F : (X, T, C1) ---->
(Y, v, C2)is () - C somewhat open function.
Proof : Let U be any nonempty open set of X. Since A is dense in X, U n A f cp. Now
Un A E TA and f is() - C somewhat open. Then there exists a nonempty open set V of Y
such that V.~ f((U n A);A )- 1). By the above result we have
(Un A);A ~ (Un A).[U n A ~ A]~ U.. Since F is an extension of f, from 1) we have
V.~ F(U.). Hence F is() - C somewhat open function.
Theorem 6.4 If (X, T, C1) and (Y, v, C2) are convex topological spaces and X = AUB where
A, B ~ X and f : (X, T, C1) ----> (Y, v, C2) is a function such that the restrictions JIA and JIB
are () - c somewhat open, then f is () - e somewhat open function.
Proof : For any nonempty open set U of X, we have U = X n U = (A U B) n U =
(A n U) u (B n U). Suppose that A n U f cp. Since flA is () - C somewhat open and
A n U(f cp) E TA, there exists a nonempty open set V of Y such that V.~ f((A n U);A)
=>V. ~ J((A n U).)[by Result 6.2] => V. ~ f(U.). This shows that f is() - C somewhat
open function.
If B n U f cp, by using f Is is B-C somewhat open, we can similarly prove that f is() -C
somewhat open.
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© Del documento, de los autores. Digitalización realizada por ULPGC. Biblioteca Universitaria, 2017