Irnat. J. M. Math. Si.
(1981)445-450
Vol. 4 No.
445
SEMI SEPARATION AXIOMS AND HYPERSPACES
CHARLES DORSETT
Department of Mathematics, Texas A&M University
College Station, Texas
(Received April 21, 1980 and in revised form September 4, 1980)
ABSTRACT.
In this paper examples are given to show that s-regular and s-normal
are independent; that s-normal, and s-regular are not semi topological properties;
and that (S(X),E(X)) need not be semi-T
s-regular,
semi-T2,
and T
0.
is compact, s-normal,
I even if (X,T)
Also, it is shown that for each space (X,T),
(S(Xs0) ,E(Xs0)) are homeomorphic, where
(X0, Q(X0)) is the T0-identification space of (X,T) and (Xs0,Q(Xs0)) is the
semi-T0-identification space of (X,T), and that if (X,T) is s-regular and
(S(X) ,E(X)), (S(XO) ,E(Xo)), and
,
then (S(X), E(X)) is semi-T
2.
KEF WORDS AND PHRASES. Smi open sets, smi topological propZies, and
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
i.
54AI0, 54B20.
INTRODUCTION.
Semi open
sets’were
DEFINITION I.i.
first defined and investigated by Levine [i] in 1963.
Let (X,T) be a space and let A -X.
Then A is semi open,
denoted by A E SO(X,T),I there exists U E T such that U GA GU.
Since 1963 semi open sets have been used to define and investigate many new
topological properties.
i
Maheshwari and Prasad
0, i, 2, regular, and normal to semi-Ti, i
[2], [3], and [4] generalized Ti,
0, 1,2, s-regular, and s-normal,
C. DORSETT
446
by replacing the word open in the definitions of
Ti,
0, i, 2, regular, and
i
Except for s-normal and s-regular, the
normal by semi open, respectively.
In this
relationships between these separation axioms have been determined.
paper, the relationship between s-normal and s-regular is determined, and semi
topological properties and hyperspaces are further investigated.
AND SEMI TOPOLOGICAL PROPERTIES.
s-REGULAR -NORMAL
2.
Maheshwarl and Prasad [4] gave an example showing that s-normal does not
That example can be combined with the following example to
imply s-regular.
show that s-regular and s-normal are independent.
Let N denote the natural numbers, let T be the discrete topology
EXAMPLE 2.1
N, let e be the embedding map of (N,T) into
on
Stone-ech
(e(N), e) denote the
(8N, W)
{If
C*(N,T)}, and
f 6
let
From
compactification of (N,T).
Willard’s book [5], (8N, W) is extremely disconnected, e(N) is open in 8N, and
BN- e(N)
B
For each p N let Np
infinite.
is
each p N, there exists a function f
then
fi
is an extension of
then
0--i
n’
0--j
#
iff i
[ #
n
m
and
iff m
an --am iff
n
n6
B x W such that (i) if
such that x
n
N} U
W be the relative topology on V, and let X
1
N, let B i
n
N,
W2
=IN
0
Xn
Bi"
s-regular,
and
C
{0 c X-
Un
V
{Yn}n N,where Yi YJ
n’{{x-I n N} b{an] n # i})
I
semi-T2,
{x
n
except for finitely many n
Then W
n
b
and
% {2,...,p},
Np,
6N,
is a base for a topology
W2
TO,
and
(X S)
N and
for all n
n
N}C, SN
n
n
nN{Un 0n e(N)}3 43N, let
D{an n N}. Since UI is
countably infinite, then U
1
i
0
he a sequence such that {a
{Xn In
m, let V
n
/
Since for
0 for all i
(2) x i
Np, (3) if i, j
i
0 is infinite, then there
], and (4) B- i
i
Let {a }
n.
N
P
n < p}.
fi-l’
{(Xn,0n )}n ; BW
exists a sequence
P
{n N
iff
i
0
and
S
IUn
a i,
on
For each
j.
W1
Yl
X,
is not s-normal since A
for all
0}, and let
(X,S)
{a
is
n
n
N } are disjoint closed sets and there do not exist disjoint
semi open sets containing
A and
C, respectively.
Semihomeomorphisms and semi topological properties were first introduced
and investigated by Crossley and Hildebrand [6].
N}
447
SEMI SEPARATION AXIOMS AND HYPERSPACES
DEFINITION 2.1. A i-i function from one space onto another space is a
semihomeomorphism iff images of semi open sets are semi open and inverses of
A property of topological spaces preserved by
semi open sets are semi open.
called a semi topological property.
semihomeomorphisms is
Example 1.5 in [6], which was used to show that normal and regular are not
semi topological properties, also shows that s-normal and s-regular are not
semi topological properties.
Clearly,
3.
0, i, 2,
i
semi-Ti,
are semi topological properties.
HYPERSPACES AND SEMI SEPARATION AXIOMS
DEFINITION 3.1.
(X,T) be a topological space, let A cX, and define
Let
S(X), S(A), and I(A) as follows:
{F S(X)
S(A)
F CA}, and
{F cX
S(X)
{F S(X)
I(A)
F
closed},
is nonempty and
}.
A
F
Denote by E(X)
the smallest topology on S(X) satisfying the conditions that if G 6 T, then
S(G) 6 E(X)
Michael
Gi
T
I(G)
and
[8] showed that for a space
Np
for all i
G
i
#
{l,J..,p}}
<G
natural numbers and
F
is
(S(X), E(X))
is semi-T
T
Np},
Since
O
0.
{<G I
(X,T), B
is a base for
Let X
E(X), where N
{F
DEFINITION 3.2
(X0, Q(X0)),
xRy
iff
where
Let
{x--
N
p
and
is the
S(X)
F c
i- Gi
and
and observed that for each space (X,T),
TO
implies
semi-T0,
then for each space (X,T),
The following example shows that (S(X),E(X)) need not
{a,b,c,d}
and T 0.
semi-T2,
{X,,{b},{d},{b,d},{a,b,d},{b,c,d}}.
and T
{a,b,c}, X
and there does not exist a semi open set containing
defined by
Gp>
P
<Gi>PiffiI
Then (S(X), E(X)) is not semi-T since
1
In Willard’s book [5],
[7].
is called a hyperspace
(X,T) is compact, s-normal, s-regualr,
if
1 even
EXAMPLE 3.1
G >
p
I
for all i
(S(), E(X))
be semi-T
Then (S(X), E(X))
E(X).
T0-identification
S(X) such that {a,b,c} #
{a,b,c}
and not
X
X.
spaces are discussed.
R be the equivalence relation on a space (X,T)
{-.
Then the
space of (X,T) is
T0-1dentification
X0 is the set of equivalence classes of R
decomposition topology on X0, which is T
0.
and
Q()
is the
C. DORSETT
448
This author
[9] used
T0-identification
spaces to show that hyperspaces of
spaces, spaces which were first defined and investigated by Davis [i0], are T I.
A space (X,T)
DEFINITION 3.3..
is
0 E T
R iff for each
0
x E 0,
and
{x) c 0.
Since T
1 implies
semi-Tl,
space is semi-T I.
then the hyperspace of each
Semi open sets were used by Crossley and Hildebran [ii] to define and
investigate semi closed sets and semi closure.
Let (X,T) be a space and let A, B c X.
DEFINITION 3.4.
Then A is semi
closed iff X-A is semi open and the semi closure of B, denoted by scl B, is the
intersection of all semi closed sets containing B.
[12] used semi closure
This author
to define and investigate semi-T
0
identification spaces.
DEFINITION 3.5.
Let R be the equlvalence relatlon on a space (X,T) defined
by xRy iff scl{x}
scl{y}.
(Xs0 Q(Xs0),
XS0
where
Then the
semi-T0-identification
space of (X,T) is
is the set of equivalence classes of R and
decomposition topology on
XS0
is continuous, closed, open, onto, and P
PS:
PI(Ps(0))
0 for all 0
(X,T)
/
is the
which is senti-T 0,
This author [13] and [12] showed that the natural map P:
natural map
Q(Xs0)
(Xso, Q(Xs0))
S0(X,T).
-1
(P(0))
(X,T)
(X0,Q(Xo))
/
T and that the
0 for all 0
is continuous, closed, open, onto, and
These results are used to obtain the
following result.
THEOREM 3.1.
(S(X0),E(X0)),
For a space (X,T),(S(X),E(X)),
and
(S (Xs0), E (Xs0)) are homeomorphic.
PROOF:
(S(X),E(X)).
Let f:
(S(Xs0),E(Xs0))
defined by f(F)
/
(S(Xo),E(X0))
P(F) and
and let
PS(F).
fs(F)
fs:
(S(X),E(X))
Then f and
fs
/
are
homeomorphlsms.
THEOREM 3.2.
S(G)
R0,G
q T, and F
S(X) such that F
i # @,
S(G) and F E l(G).
PROOF:
A
If (X,T) is
S(G).
Since S(G) c S(G), which is closed, then S(G) c S(G).
Let
p
<Bi>i= 1
B
such that A
P
<Bi>i= 1.
Then A c G and
Let
then
449
SEMI SEPARATION AXIOMS AND HYPERSPACES
# A
Bic
For each 16 N
i
--N P{xi }
S(G)
Bi
G
P
let x
E S(G)I
G
i
.Bi.
e
<Bi>i=l.
which implies G
Then _{x cG
i}
Thus AE
’(G)
and
iB i # @
for all i
Np.
.RBi for all iE NP and
S(G) c S(G), which implies
S(G).
Let
each i 6
m
<Ui>i= 1
E 8 such that F
Nm let Yi Bi"
THEOREM 3.3.
PROOF:
say B- A #
W
Np,
for all i
.
en } U
Let x E B- A.
and A cW.
Then Fc
im }
If (X,T) is s-regular and
Let A, BE S(X)
such that x E 0
<Ui>=I.
,
such that A # B.
im UIE
I(G) C,
m
<Ui>i. 1.
and
Hence
then (S(X),E(X)) is semi-T
2.
Then A- B #
or B
A # @,
Then there exists disjoint semi open sets
Let U, V ET
0 and
such that Uc 0 cU and V cW cV.
Then I(U) and S(V) are disjoint open sets, B E I(U), and A E S(V)
implies S(V) b {A} and
T
S(V), which
I(U) U {B} are disjoint semi open sets.
Maheshwari and Prasad [4] showed that every s-normal
space is s-regular.
This result can be combined with Theorem 3.3 to obtain the following corollary.
COROLLARY 3.1.
If (X,T) is s-normal and R0, then (S(X),E(X)) is semi-T
2.
C. DORSETT
50
REFERENCES
i.
LEVINE, N., Semi Open Sets and Semi Continuity in Topological Spaces,
Amer. Math. Monthly, 7_0 (1963), 36-41.
2.
MAHESHWARI, S. and PRASAD, R., Some New Separation Axioms, Ann. Soc. Sci.
Bruxelles, 89 (1975), 395-402.
3.
MAHESHWARI, S. and PRASAD, R., On s-Regular Spaces, Glasnik Mat. Ser. III,
i0 (30) (1975), 347-350.
4.
MAHESHWARI, S. and PRASAD, R., On s-Normal Spaces, Bull. Math. de la Soc.
Sci. Math. de la R. S. de Roumanie, T 22(70) (1978), 27-30.
5.
WILLARD, S., General Topology, Addlson-Wesley Publishing Company, 1970.
6.
CROSSLEY, S. and HILDEBRAND, S., Semi-Topological Properties, Fund. Math.,
74 (1972), 233-254.
7.
FRINK, 0., Topology in Lattices, Trans. Amer. Math. Soc., 5_i (1942), 569-582.
8.
MICHAEL, E., Topologies on Spaces of Subsets, Trans Amer. Math. Soc., 7__1
(1951) 152-182.
9.
DORSETT, C., T^-Identification Spaces and Hyperspaces, Ann. Soc. Sci.
9__1 (1977), 200-206.
Bruxelles?
i0.
DAVIS, A., Indexed Systems of Neighborhoods for General Topological Spaces,
Amer. Math. Monthly, 68 (1961), 886-893.
ii.
CROSSLEY, S. and HILDEBRAND, S., Semi-closure, Texas J. Science, 22 (1970),
99-112.
12.
DORSETT, C., Semi-T^-Identification Spaces, Semi-Induced Relations, and
Semi Separation Axioms, Accepted by the Bull Calcutta Math. Soc.
13.
DORSETT
C., To-Identification Spaces and R1 Spaces, Kyungpook Math. J.,
(1978), 167-174.
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