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Int Jr. of Mathematics Sciences & Applications
Vol.3, No.1, January-June 2013
Copyright  Mind Reader Publications
ISSN No: 2230-9888
www.journalshub.com
A VIEW ON NEW HYPERSPACE TOPOLOGY
VIA-SEMI OPEN SETS
G. Vasuki
[email protected]
E. Roja and M.K. Uma
Department of Mathematics,
Sri Sarada College for Women,
Salem - 636 016
Tamil Nadu.
Abstract
In this paper the concepts of *-semi open set, *-semi closed set, a new topology *(X), H-semi
closed, semi-Urysohn space are introduced. In this connection different properties of *(X) are
investigated.
Keywords
*-semi open, *-semi closed, new topology *(X).
2010 AMS Mathematics subject classification : Primary : 54A05, 54A10, 54A20
1. Introduction
Hyperspace
topology
was
first
initiated
and
extensively
studied
by
F. Hausdorff. Hyperspace is the collection of certain subsets of a topology space, equipped with a
suitable topology. In the study of hyperspace topology, the first step towards topologizing a collection of
subsets of a topological space X was taken by Hausdorff [2], after that many famous mathematicians
have tried in multi various ways to topologize suitably different collections of subsets of a topological
space, some of them are Kuratowski, Vietoris, Michael, McCOY and Fell. In this paper the concepts of
*-semi open, *-semi closed, a new topology *(X), H-semi closed and semi-Urysohn are introduced.
Some interesting properties of *(X) are discussed.
2. Preliminaries
Definition 2.1 [3]
A point x  X is said to be a -contact point of a set A  X if for every neighbourhood U of x,
clX U
 A  .
The se of all -contact points of a set A is called the -closure of A and denote this set by -clX
A. A set A is called -closed if A =  clX A.
A set A is called -open if X \ A is -closed.
Remark 2.1
41
G. Vasuki, E. Roja and M.K. Uma
The collection of all -open sets in a space X forms a topology on X. In this connection (X) a
new topology T is defined as (X) = {A  X; A   and A is
-closed in X} where X is a topological space.
Definition 2.2 [3]
A T2-space X is called H-closed if any open cover u of X has a finite proximate subcover, i.e., a
finite subcollection uo of u whose union is dense in X.
A set A  X is called an H-set if any cover {U :   } of A by open sets in
X has a finite subfamily { U α : i = 1, 2, …n} such that A 
i
n

i 1
clX U α
i
Definition 2.3 [1]
A topological space is a T0-space iff for each pair x and y of distinct points, there is a
neighbourhood of one point to which the other does not belong.
Definition 2.4 [1]
A topological space is a T1-space iff each set which consists of a single point is closed.
Definition 2.5 [1]
A topological space is a T2-space iff whenever x and y are distinct points of the space there
exists disjoint neighbourhoods of x and y.
Definition 2.6 [1]
A set A is dense in a topological space X iff the closure of A is X.
3. *(X) with A new topology
Definition 3.1
Let x be a point in a topology space (X, T). A set U in X is said to be semi neighbourhood of X
if there exists a semi open set G in X such that x  G  U.
Definition 3.2
Let (X, T) be a topological space. Let A be a subset of (X, T). The intersection of all semi
closed sets containing A is called the semi closure of A and is denoted scl(A).
That is, scl(A) =
 { B  P (X) / A  B, B is semi closed set in X }
Definition 3.3
A point x  X is said to be a *-semi contact point of a set A  X if for every semineighbourhood U of x, sclXU
 A  .
The set of all *-semi contact points of a set A is called the *-sclosure of A and denote this set
by *-sclXA.
A set A is called *-semi closed if A = *-sclXA. A set A is called
*-semi open if X \ A is *-semi closed.
Remark 3.1
The collection of all *-semi open sets in a space X forms a topology
on X.
Definition 3.4
Let X be a topological space, *(X) is said to be new topology if
*(X) = {A  X; A   and A is *-semi closed in X}
*(X) a new topology on T is denoted by (*(X), T)
Definition 3.5
A topological space is a sT2 space iff whenever x and y are disTinct points of the space there
exists disjoint semi neighbourhoods of x and y.
Definition 3.6
A set A is semidense in a topological space X iff the semiclosure of
A is X.
42
A VIEW ON NEW HYPERSPACE TOPOLOGY VIA-SEMI OPEN SETS
Definition 3.7
A ST2-space X is called H-semi closed if any semi open cover U of X has a proximate semi
subcover, that is, a finite subcollection Uo of U whose union is semi dense in X.
A set A  X is called a H-semi set if any cover {U :   } of A by semi open sets in X has a
finite subfamily
n
{ U α : i = 1, 2, …n} such that A 
 sclX Uα
i
Definition 3.8
On *(X),
define
a
topology
i 1
as
follows:

+
W = {A  * (X) : A  w} and W = {A  * (X) : A
Sθ* =
Consider

{ W
i

For
each

W
X,
let
W  }.
: W is semi open in X}

+
{ W
: W is *-semi open in
Sθ* form a sub-basis for some topology on *(X) which denote by
X and X | W is an H-semi set}. Then
T.
Proposition 3.1
Let P1, P2, … Pn be subsets of X. Then




 Pn


+

(a)
P1  P2  P3
(b)
Let P1, P2, …… Pn be *-semi open sets and X \ Pi is an H-semi set for
…….
i = 1, 2, … n. Then (P1
Proof
(a)

Let A 



P2



……

P2
P n) + 

……
…….
 Pn

……

Pn, i.e., A  (P1


……

…….
P2

 Pn
 (P1
P2
Pn) .
Sθ*
P1  P2  P3
i = 1, 2, ….n. Hence, A  P1
P1  P2  P3

= (P1
. Then A  * (X) with A  Pi for each



P2

…




(b)
…

P2

…
Pn) . Therefore






P1  P2  P3

Pn
. Thus
…

Pn

P1  P2  P3
. Therefore, (P1
…
Since each Pi is *-semi open for i = 1, 2, ….n, P 1


Pn
P2
= (P1


P3

P2
P2


+
Pn)
Pi  ,
Pn. Hence B  Pi for each i = 1, 2, ….n, i.e., B 
i = 1, 2, …n. That is B 
P1  P2  P3
……
Pn) +.
Conversely, let B  *(X) such that B  (P1
B  P1
+

P2
…
i.e.,
for each
+

P n) 
+
…  Pn) .
 …  Pn is also *-semi
 (X \ P2)  …. 
open. Now, (X \ P1  P2  ……  Pn) = (X \ P1)
(X \ Pn). Since each (X \ Pi) is H-semi set for i = 1, 2, ….. n and union of finitely many H-semi sets is an
H-semi set. Therefore, X \ (P1


P2

……

Pn) is an H-semi set. Hence (P1

P2

……

P n) +
Sθ* .
Remark 3.2
Using the above proposition we can say that any basic semi open set in the above defined
topology
is
of
the
form




P1  P2  ..... Pn  Po
where
P1, P2, ….. Pn are semi open in X, Po is a *-semi open set with X \ Po an H-set. We may also choose
each Pi  Po, for i = 1, ….. n in such a basic semi open set.
Definition 3.9
A topological space is a sTo space iff for each pair x and y of distinct points, there is a semineighbourhood of one point to which the other does not belong.
43
G. Vasuki, E. Roja and M.K. Uma
Proposition 3.1
(*(X), T) is always sTo.
Proof
Let A, B  *(X) be such that A  B. If A  B then A  (X \ A)


B

(X
\
B)

=

B

(X
and

 B then A  (X \B)    A  (X \ B) .
B  (X \ A) . Now (X \ A) is semi open in (*(X), T). If A
Also

\
B).
Since
B
is
*-semi
closed,
X \ B is *-semi open in X. Hence (*(X), T) is To.
Definition 3.10
A topological space is a sT1 space iff each set which consists of a single point is semi-closed.
Proposition 3.2
X is sT2 iff {a} is *-semi closed for each a  X.
Proof
Let X be sT2 and a  X. We prove that X \ {a} is *-open. Let x  X \ {a}. Since X is ST2 there
exists two disjoint semi open neighbourhood U, V of x and a respectively. Thus U

V =   sclX U

V
=


x

U

sclX
U

X
\
{a}.
So
X \ {a} is *-semi open which implies that {a} is *-semi closed.
Conversely, let {a} be *-semi closed for all a  X. Let x, y  X be such that
x  y. Since {y} is *-semi closed, there exists a semi open neighbourhood U of x such that y  sclX U
and hence y  X / sclX U. But x  U and U

(X / sclXU) = . Hence
X is sT2.
Proposition 3.3
(*(X), T) is sT1 if X is sT2.
Proof
Let A, B  * (X) be such that A  B. Without loss of generality let A

B. Then A
 (X \ B)  

 A  (X \ B) which is a semi open set in (*(X), T) since (X \ B) is *-semi open. Also there exists a
 A such that a  B. Then B  (X \ {a})+. Since X is sT2, by proposition 3.2, {a} is *-semi closed and
hence X \ {a} is *-semi open. Also {a} is a H-semi set for each a  A. Hence (X \ {a}+ is semi open in
(*(X), T). Thus (*(X), T) is sT1.
Definition 3.11
Let < s,  > be a semi Urysohn space iff x, y  s with x  y implies that there exists U, V  
with x  U, y  V and scl (U)

scl (V) = .
Proposition 3.4
(*(X), T) is sT2 if X is semi Urysohn and H-semi closed.
Proof
Let A, B  *(X) be such that A ≠ B. Without loss of generality let A

B. Then there exists a
 A such that a  B. Since B  *(X), a  B = *-sclXB. Thus there exists a semi neighbourhood U of a
such that sclXU
semi
a H-semi
A



set.
B =   B  X \ sclXU. Since X is semi Urysohn and H-semi closed, sclXU is *Let
V
closed
= X \ sclXU.

Then
+
V
U    A  U and B  V . Now, we show that U
(X
(X \ sclXU)
+
\

sclXU) .
Then
P

U

is



and
*-semi
44
set
in
X.
(X \ sclXU) = . If possible, let P  U
and
U   which is a contradiction. Hence (*(X), T) is sT2.
Proposition 3.5
open
+
also
Thus
P

X
\
sclXU


A VIEW ON NEW HYPERSPACE TOPOLOGY VIA-SEMI OPEN SETS
Let P1, P2, …. Pn be semi open sets in X and Po be *-semi open set in X. Then in (*(X), T),
sclX ( P1
 P2  .....  Pn Po ) = (sclXP1) 

(sclXP2)


……

+
(sclXP0) , provided X is semi
Urysohn and H-semi closed.
Proof
Let A  (sclXP1)

A
sclXPo or A




(sclXP2)


……


+
(sclXP0) . Then either A

sclXPi = , for some i, where 1  i  n. If A

(X \ sclXPo)    A  (X \ sclXPo) . But (X \ sclXPo)

(sclXPn)

Pn Po . Now, if A


P1  P2  ..... 



sclXPo, then
L = , the empty set in *(X) where L =
sclXPi = , for some i, then A  X \ sclXPi  A  (X \
+
sclXPi) . Since X is semi Urysohn and H-semi closed, sclXPi is *-semi closed and H-semi set. So (X \
sclXPi)
+
is
(X \ sclXPi)

+
open

1




)  (sclXP1)


(sclXP2)

 P2  ..... 

 …. 
(sclXPn)
Now,
Pn Po ). Therefore,

L = . This shows that A  scl*(X) ( P
 P2  .....  Pn Po
scl*(X)( P1
*(X).
in


(sclXPo)
+
………. (3.5.1)
Now,
let



A
S1  S2  ..... 
V =

(sclXP1)

Sm So

…..  (sclXPn)
(sclXP2)

(sclXPo)+
and
be a semi open neighbourhood of A in *(X). Then
S1, S2, …. Sm are semi-open and So is semi-open in X with X | So H-semi set such that Si  So, i = 1, 2, …
m. And A

sclXPj   for all j = 1, 2, 3, ….n which implies that there exists aj  A
…..n. Also A  So. Therefore So being a semi open neighbourhood of aj, So
implies
that
there

xj  So
bi  A


,
Now
i
=
1,
2,

Si
,
i


=
1,
+

2,
which

….m
implies
that
there

Po
exists



 P2  .....  Pn  Po
and
B

So.
Also
B

Pj

,
L. Hence A  sclxL. So,

(sclXP2) …..  (sclXPo)  scl ( P1
From (3.5.1) and (3.5.2) we get
scl ( P1
Si  , i = 1, 2, …. implies that there exists
….m
j = 1, 2, ….n and B  Po. Therefore B  V
(sclXP1)
Pj   for j = 1, 2, ….n
exists
Po, i = 1, 2, ….m. Let B = {x1, x2, … xn, w1, w2, … wm}. Since X is sT2, B is semi-closed.

B


sclXPj, j = 1, 2,
Si, i = 1, 2, 3 …. m. Also A  sclXPo. Therefore, as Si are open neighbourhoods of bi, Si

wi  Si
Pj, j = 1, 2, ….n. Now A


) = (sclXP1)



 P2 .....  Pn Po )


(sclXP2)

…… (3.5.2)
 ….. (sclXPo)+
Proposition 3.6
(*(X), T) is H-semi closed if X is semi Urysohn and H-semi closed.
Proof
Let {Yi} be a universal net of elements of *(X). Define Z = {x  X : for each semi open

neighbourhood U of x, {Yi} is eventually in {(sclXU) }. Choose yi  Yi. Then {yi}is a net in X which is
H-semi closed and sT2. Hence {yi} has a *-converget subnet {yni} say *-converging to y. Then for any

semi-open neighbourhood W of y, {yni} is eventually in sclXW, ie, {Yni} is eventually in (sclXW) and

hence {Yi} is eventually in (sclXW) because of the universality of {Yi}. Thus yZ and Z  . To show
that Z *(X), let {x} be a net in Z *-converging to xX. Let U be an arbitrary semi open
neighbourhood
of
x.
Since
X
is
H-semi
closed
and
semi-Urysohn,
X is almost semi regular. Hence there exists a semi open neighbourhood V of x such that x V  sclXV
45
G. Vasuki, E. Roja and M.K. Uma

sintX
(sclX
(U)).
Since
{x}
*-converges
to
x,
there
exists
a
o   such that x  sclXV  sintX (sclX(U)), for all   o and since x  z, {Yi} is eventually in

(sclXU) .
Hence

x
*-converges to Z in T. Let
of
Z
bj  Z
in

T,
that
is,
Z,
ie.,


Z

B1  B2  ..... 
Z  Bi  
*(X).
Now,
to
show
that
{Yi}

Bn  Bo be an arbitrary semi open neighbourhood

for
all
i
=
1,2,…n
and
Z

Bo.
Let
Bj for all j = 1,2,…n. Since Bj is a semi-open neighbourhood of bj, bj Z  {Yi} is eventually

in (sclXBj) for j = 1,2, …n. Therefore, {Yi} is eventually in (SclXB1)


 (sclXB2) 
….


(sclXBn)
+
Now, it is sufficient to show that {Yi} is eventually in (sclXBo) . Since {Yi} is a universal net, either {Yi}
is
eventually
in


in *(X) \
B0 or
B0 .
such that Yi *(X)\

B0 ,

If {Yi} is eventually in *(X) \
B0 ,
then there exists io
for all i  io, i.e. Yi  (X \ Bo)  , for all i  io. Choose
Zi  Yi  (X \ Bo) for i  io. Then X \ Bo being an H-set, {zi} has a * convergent subnet {zni} say *converging to Z. Clearly z  X \ Bo. Then for any semi open neighbourhood W of z, {zni} is eventually
in
sclXW
ie.,
{Yni}

is
eventually
in

(sclXW) and hence {Yi} is eventually in (sclXW) by the universality of {Yi} which implies that z  Z
 zZ
 (X \ Bo) which contradicts the fact that Z  Bo. Hence {Yi} is eventually in B 0 , that is in
+
(sclXBo) .
(sclXB1)

Thus
{Yi}


is
 (sclXB2)  …  (sclXBn)  (sclXBo)
+
eventually

= scl*X ( B1
in

Bn  Bo ) which


 B2  ..... 
implies that {Yi} *-converges to Z in T. Hence (*(X), T) is H-semi closed.
Definition 3.12
A topological space is semi-compact iff each semi-open cover has a finite subcover.
Proposition 3.7
If X is sT2 and (*(X), T) is semi compact, then X is semi-compact.
Proof
Let {U :} be a semi open cover of X. Let x X. Then xU for some . Since X is


sT2, {x} is *-semi closed, ie., {x} *(x) and so, {x} U  , for . Hence { U  :} is a T-open
n
cover of *(x). Since (*(x), T) being semi compact, *(x) =
n
=

Ui

i1
,

{y}  U m
i.e.

Ui

i1

. Let y X, Then {y}*(x)

for
some
n
m where 1  m  n, that is y U m . Hence X =
U i . Then X is semi-compact.

i1
Proposition 3.8
If X is sT2 and (*(X), T) is semi-Urysohn, then X is semi-Urysohn.
Proof
Let x, y  X be such that x  y. Now, X being T2; {x}, {y}  *(X) and
{x}  {y}. Since (*(X), T) is semi-Urysohn, there exists a semi open neighbourhood







U1  U2  .....  Un  Uo of
V1  V2  .....  Vm
{x}
and
a


 Vo of {y} such that scl*(x) ( U1
46
semi

open
neighbourhood


 U2  .....  Un  Uo )  scl*(x)
A VIEW ON NEW HYPERSPACE TOPOLOGY VIA-SEMI OPEN SETS




( V1
 V2  .....  Vm  Vo
) =  where U1, U2, … Un, V1, V2, …. Vm are semi open in X; Uo, Vo
are
*-semi
with
open
in
X
X
\
Uo,
X


 V1



 V2  .....  Vm  Vo
H-sets,
 U2  ..... 
U1  U 2  .....  U n  U o =
{y}
Vo,

i = 1, 2, … n, Vi  Vo for i = 1, 2, … m. Now, {x}  U1

\

Ui
Uo
U1  U 2  .....  U n ,
implies
that
y

for

Un  Uo implies that x

and



V1  V2  .....  Vm  Vo =

V1  V2  .....  Vm . We have to prove that sclx ( U1  U 2  .....  U n ) 
sclx ( V1
sclx
W
 V2  .....  Vm )
= . If not, let z  sclx ( U1
 V2  .....  Vm ).
( V1
Then
for
each
semi
open
 U 2  .....  U n ) 
neighbourhood
W
of
Z,
 U1  U 2  .....  U n  U o   and W  V1  V2  .....  Vm  Vo  . Since for
pX, p W
{p} W
that
{z}

 U1  U 2  .....  U n  U 0 implies that




 U1  U2  .....  U n  U0



, hence, W

U1  U2  ….
Un  U0   implies


W  V1  V2  .....  Vm  Vo  . Then


scl*x( U1



 U2  .....  Un  Uo )  scl*(x)
semi-open
neighbourhood
V1  V2  .....  Vm
 U 2  .....  U n )  sclx
( V1

( V1
a contradiction. Hence there exists a semi open neighbourhood U1
( U1



 U 2  .....  U n
of

 V2  .....  Vm  Vo
y
 V2  .....  Vm )
such
of x and a
that
=
)
sclx
.
Thus X is semi-Urysohn.
Definition 3.13
A space X is locally *-H if X contains a base B for its topology such that for each B  B, sclx
B is an H-semiset and *-semi closed.
Proposition 3.9
If X is H-semi closed and semi-Urysohn, then X is locally *-H.
Proof
Let B be a base for the topology of X. Then for each x  X, there exists a basic semi open set B

B,
such
that
x

B.
Now,
B
being
semi
open,
sclx B= *-sclx B. Also, X being H-semi closed and semi-Urysohn, sclx B is
semi-*-closed and an H-semi set since *- closed subset of an H-semi closed space is an H-semi set.
Hence
B
is
the
required
base
for
X
such
that
for
each
B  B, sclx B is an H-semi set and semi *-closed. Hence X is locally *- H.
Proposition 3.10
If X is sT2, locally *-H and (*(X), T) is H-semi closed, then X is
H-semi closed.
Proof
Let B be a base for the topology of X such that for each B  B, sclXB is a
*-semiclosed and H-semi set. Let U = {U :   } be a semi open cover of X. Without loss of
generality, assume that U belongs to B. We have to prove that there is a natural number n and there
exists
1,
2,
……n


such
that
n
X = sclX
(  Uα ) .
i1
i
If A  *(X), then A is a subset of X and intersects a U; so,
47
G. Vasuki, E. Roja and M.K. Uma
A


U α . Hence { U α
:   } is a semi open cover of *(X). Since *(X) is H-semi closed there exists
a finite proximate subcover of *(X), that is, *(X) = scl*(X) (
2,
n
……

.
We
n

U α ) for a natural number n and some  ,

i
i1
1
have
to
n
X = sclX
(  Uα ) .
i
i1
prove
that
n
If it is not the case, then there is x  X \
(  sclX U α j)
= W. But
j 1
W is *-semi open set and X \ W is an H-semi set. Since X is sT2, {x} is
*-semi closed, so {x}  W+. On the other hand, there is i  {1, 2, ….n} such that
{x}  scl*(X)
F
 Uα
i

Uα
i
. Therefore W
 , which means that W
+

 Uα
 Uα
i
+
i
 . Let F  W

 Uα
i
. Thus, F  W and
  which contradicts the definition of W. So, X must be
n
covered by sclX
(  Uα ) .
i1
i
REFERENCE
1.
John L. Kelly, General topology, 1955 Springer-Verlag Newyork Berlin Heidelberg.
2.
F. Hausdorff, Mengenlehre, 3rd Ed. Springer, Berlin, 1927.
3.
N.V. Velicko, H-closed topological spaces, Amer Math. Transl., 78
103-118.
48
(1968),