Internat. J. Math. & Math. Sci.
(1996) 67-74
VOL. 19 NO.
67
ON LOCALLY s-CLOSED SPACES
C. K. BASU
Depa[tmcnt ot Pure MathemdtCs
University of Calcutta
35, Bailygunge Circular Odd
Calcutta
700 019
(Received October 12, 1992 and in revised form July 23, 1993)
ABSTRACT.
closed
the
in
spaces have been
c{osednss
papz,
prent
seml-rgulat
is
concepts of
the
lt[oduced
and
propetty;
s-closed
chardCt[Ized.
the
We
concept ot
loCd[ly s-
sub-spaces,
local
that
seen
have
s-8-closed
s-
has been
mapping
introduced here and the following impo[t=nt ptopertles are establlshed
Let
X --)
s-8-closed
it
iS
completely
q -space, then, X
g
It
(b)
f
is
(Malo and
s-set
WONDS
AND
Is
(Ary and Gupta
continuous
8i)
Nolrl
polnt
ad Y
locally compact
is
IGanguly and
{5)
BdSU
and X
locally compact T-
s a
locally s-closed.
PHRASES.
s-closed
subspac,
-contlnuous and completely
mapping,
|i])
locally s-closed.
Is
-conttnuous
space, then,
KEY
sur3ectlon wlth
Then
Inve[ses.
(d)
be an
locally
s-set,
continuous mapplng,
s-closed,
s-8-closed
regular open set, s-0-open
set, local compactness.
1991 AMS SUBJECT CLASSIFICATION CODES. 54D99, 54C99.
I.
S-closed spaces
iNTRODUCTION.
(Thompson
L14J)
and
s-closed
[8J) spaces orlglnated from almost compact spaces by the use of
and Nolr
(Mayo
semi-open
sets as
Introduced by Levlne [7. Ganster and Reilly [6] had shown, towards the dlstlnction
between these concepts,
that every
nfnte topological space can be embedded as a
closed connected subspace of an S-closed space whlch is not an s-closed space. Nolrl
[13] further generalzed S-closed spaces to locally S-closed spaces,
we
generalize
subspaces.
spaces
s-closed
mportant
Certain
have
spaces
also
been
to
locally
s-closed
characterzatlons
established,
and
s-0-closed
spaces
and
propertles
<,f
mapping
s
in
this paper
study
s-closed
locally s-close
ntroduced
and
characterized and we have seen, under certain condltlons on the domain and co-domain
spaces,
that
an
contlnuous and
s-0-closed
mapping
would
be
a
continuous
mapping.
-contlnuous mappings were introduced respectlvely by
[I and Ganguly and Basu [5i; by the help of these
mappings
we
Completely
Arya and Gupta
have been able to
establish certain properties which corelate locally compact T -spaces wth
s-closed spaces.
locally
C.K. BASU
68
Throughout
topologlcal
the
paper,
present
A subset A of
space.
regular
int(A))
denotes the closure (resp.
to
be
[7]
semi-open
if
int(cl(A))=A
there
if
complelnent of a seml-open set
collecton
all
of
set
such
0
(resp.
whole space X these wlll be denoted by SO(X)
to
be
sclU A#. Snce, for a
of
X
said
Is
[8]
s-0-cluster
to
be
O Acl(O).
that
The
s the ntersectlon of
called a seml-regular set
is
seml-open
semi-regular,
contalnlng a pont x of X wll be denoted by SO(x)
sad
(resp.
cl(A)
where
(Crossley and Hildebrand [3]). A set A whlch is
both semi-open as well as seml-closed
The
polnt
of
seml-open set
a
(Malo and Nolrl
regular
sets
open)
(resp. SR(x), RO(x)) and for the
(resp. SR(X), RO(X)). A polnt x of X
subset
A
X
of
If
every
for
U (SO(x),
U, sclU is a semi-regular set [8], a polnt x
an s-0-cluster pont of A ff R A,
for all
SR(x).
R
The
collection of all s-0-cluster points of A will be denoted by s-O-clA ([A]
A set A is s-0-closed if
short).
s-O
A complement of an s-0-closed set
A=[A]s_ 0
called an s-0-open set. For a space (X,T), RO(X,T) is
coarser
than
T
and
(X,Ts)
is
called
the
a base for a topology T
semi-regularizatlon
topologlcal property P xs said to be semi-regular
property
space of
set),
A
then X
Nearl
compact
sclU
on X
(X,T).
A
A subset A
(X,Ts).
(resp. S-closed (Nolrl [II])) relatlve to X or slmply an s-set
(resp. S-set) if every cover
such that
S
for
if whenever a space (X,T)
possesses that property P so does its seml-regularzation space
of X Is s-closed [8]
a
called seml-closed (Crossley and Hildebrand 13]).
semi-closed sets containlng A
[8]).
mean
shall
we
of A. A subset A of a space X is sald
The semi-closure of a subset A of a space, denoted by sclA,
all
X
by
cl(Int(A))=A),
open
an
slmply
space s said to be regular open
(resp.
nterior)
exlsts
is
or
(X,T)
a topological
(resp.
closed)
by
of A by sets of SO(X)
(resp. A
C
is called s-closed
(NC-set
In case A
clU).
[8]
(resp.
admits a finite
X and A
U
1
n
(resp. S-
[14]). A subset A is called
S-closed
(Carnahan [2]), for short) if every cover
open sets of X has a finite subfamily U
subfamlly
an s-set
Is
of A by means of
AU
(say) such that
intclU.
Clearly every s-set (resp. compact) set, is an NC-set, but not conversely. A subset
A of a space X is said to be an O-set (Noiri [10]) if Aint(cl(int(A))).
2.
s-CLOSED SUBSPACES.
At the very outset, an example is glven to assert that,
every set, s-closed relative to X, is not necessarily an s-closed subspace of X.
EXAMPLE
I.
Every
countable
set
in
an
uncountable
set
X with
co-countable
topology T s s-closed relative to (X,T), but is not even an S-closed subspace.
DE’INITION i. A subset A of X s said to be pre-open (Mashour et al. [9])
A intclA. This collection includes all open sets and, even more, all -open sets.
LEMMA i. (See Dorsett [4]) Let (X,T) be a topological space and let A be preopen set in (X,T), then
SR(A,TA)=SR(X,T)
A, where T is the subspace topology on A.
A
THEOREM i. A pre-open set A of X is s-closed as a subspace iff it is s-closed
relative to X.
PROOF. Let A be s-closed relative to X and also let
A by semi-regular sets of the subspace A.
regular set Um in X, for each
A
is
A
s-closed
U,
to
which shows that
m%Iub_space
as
relative
X,
I, such that
there
exlsts
A (U A)
I0
IV
m
Then by Lemma I,
i.e.,
U A.
V
a
finite
A
Therefore,
subset
V
I0
m
I
}
be a cover of
there exists a semi-
I
o
AU.
of
Therefore, A
I
such
is
Since
that
s-closed
LOCALLY s-CLOSED SPACES
let A be s-closed as a subspace.
Conversely,
o
semi-regular sets
there exists a
U
Then A
X.
flnlte subset I
(V
of
o
Let
A).
69
Vm
U
I such that A
(Vm A),
THEOREM 2. Let B be a pre-open set
tzl
which shows that
I0
(
Va. Therefore A is s-closed relative to X.
A
be a cover of A by
I
Since A as s-closed as a subspace,
(X,T). Then a subset A of B as s-closed
relative to the subspace B iff A as s-closed relatlve to X.
PROOF. The proof follows by Lemma I.
COROLLARY I. Let A and B be open sets of a space X such that A B. Then A as
an s-cloed subspace ot B ff A as an s-closed subspace of X.
PROOF. Applyng Theorem 1 and Theorem 2, we get the result.
DEFINITION 2. Let
for a topology called T
(X,T) be a topological space, then SR(X,T} forms a sub-base
SR
-topology on X.
LEMMA 2. A subset A of a space (X,T} is s-closed relative to (X,T) if A is
compact n
(X,TsR)Let A be s-closed relatlve to (X,T}. Then every cover of A by sets of
PROOF.
SR(X,T}
has a finite subcover.
But SR(X,T)
sub-basic open cover of
base theorem A is
forms a sub-base for (X,’ ). So every
SR
a
flnte
has
subcover. Therefore by Alexander sub-
(X,TsR)
n (X,TsR)compact in (X,TsR)
comDact
Coverse[y, if A is
then every sub-basic open cover has a finite
subcover. So every cover by sets of SR(X,T} has a finite subcover. Therefore A is sclosed relative to (X,T).
THEOREM 3.
Let B be a T -closed set in X and let A be any subset of X which
SR
s s-closed relative to (X,T). Then AB is s-closed relative to (X,T}.
{U
TsR-Open
PROOF. Let
{X-B)
so,
as a
there
exists
imples that
A
a
6I
TsR-Open
be a
AnB.
cover of
Then clearly
cover of A. By Lemma 2, A as compact relative to
flnlte
ABC
subset
Ua
I
o
Therefore
of
1
AB
such
that
s compact
A
in
I0
s s-closed relative to (X,T).
e Ua} U
{ Ua I}
(X,TsR}; and
(X--B)
which
o
(X,T }. Then by Lemma 2,
SR
COROLLARY 2. If B is regular open or regular closed and A as any subset of X
which s s-closed relative to X, then
A5
as s-closed relative to X.
Since every regular closed or regular open set is sem-reular,
PROOf.
the
corollary follows from Theorem 2.
COROLLARY 3. If X is an s-closed space and A is a regular open set of X, then
A is an s-closed subspce of X.
PROOF. The proof follows from Theorem 1 and Theorem 3.
COROLLARY 4. If A is s-closed open subspace of X and
of X, hen
AO
PROOf.
is a regular open set
is an s-closed subspace of X and (hence of A and B).
The
proof
follows
from
Corollary
2
and
Theorem
1
and
second
prt
follows from Corollary 1.
THORFq
n
then
i
1,2,...,n are s-sets i.e.,
i
A. is s-closed relative to X.
4.
If A
s-closed relative to X.
i=l
PROOF.
Straightforward.
THEOREM 5. Let X be an s-closed spce and let A be a closed set of X and let
frontier of A,
relative to X.
denoted
by Fr(A),
be
s-closed
relative
to X.
Then A
is s-closed
70
C.K. BASU
PROOF.
Since X
intA is s-closed
by Corollary 3 and Theorem I,
s-closed,
is
closed set. Sance A=intA UFr(A), by Theorem 4, A as s-
relatave to X whenever A as
closed relative to X.
LOCALLY s-CLOSED SPACES
3.
DEFINITION 3. A space X as saad to be locally s-closed aff each polnt belongs
to a regular open neaghbourhood (nbd. for short) which is an s-closed subspace of X.
locally s-closed space. However,
REMARK I. Clearly every s-closed space is
the
converse
is
because
ingeneral,
true,
not
any
uncountable
set
with
discrete
topology is locally s-closed but not s-closed.
A topologacal space
THEOREM 6.
is locally s-closed iff for each point
(S,T)
x { X, there exists a regular open set U containing x such that U is locally s-closed.
At farst we prove that if A is a regular-open set in (X,T)
PROOF. Sufficaency
then every regular-open set in the subspace
Let VCA be regular-open an the subspace
intx(AClxV)
ant cl V Cint cl A
X X
X X
of X. Then,
antAClAV intA(AClxV)
ant cl V
X X
X
Therefore V is regular open an (X,T).
A).
poant
(A,TA).
Then V
Aint cl V
intxA_intxcl XV
by
X
hypothesis,
there
exists
(X,T).
is also regular-open in
A)
(A,T
a
VA
(as
implies
Now let x be any
(X,T)
of
U
set
regular-open
containlng x such that U as locally s-closed. Then there exists a regular open set V
in U such that x
EV
and V is an s-closed subspace of U.
Therefore V as a regular-
open set in (X,T) and by Corollary I, V is s-closed subspace of X. Therefore (X,T)
Is
locally s-closed.
The proof as straightforward.
Necessity
THEOREM 7. Let (X,T) be a topological space. The following are equivalent
(i)
X is locally s-closed;
(ii)
every polnt has a regular-open set whach is s-closed relative to X;
(iii)
every point x of X has an open nbd U such that int cl U is s-closed
X
X
relative to X;
(iv)
every
x
point
of
X
has
an
open nbd U such that sclU is s-closed
relative to X;
(v)
for every point x of X, there exists an
-open
set V contaanang x such
-open
set V containing x such
that sclV is s-closed relatave to X;
(vi)
for every point x of X, there exists an
that int cl V is s-closed relative to X;
X
(vii)
X
for each x
X,
there exists a pre-open set V containing x such that
sclV is s-closed relative to X;
(viii) for every x of X, there exists a pre-open set V containing x such that
(ix)
int cl V is s-closed relative to X;
X X
for every x X, there exists a pre-open set V containing x such that
--
intxclxV
PROOF.
(i)
is an s-closed subspace of X.
(ii)
Follows
regular-open set is pre-open set.
(Maio and Noiri [8]).
forward
(iv)
(vi)
-->
(v) is evident, since every open set is
(vii),
fact
that every
sclU
intclU
-open.
(vii) --) (viii) and (viii) --) (ix) are straight-
because of the facts that every
open iff sclV
and from the
Follows from the fact that for an open set U,
(iii) ---) (iv)
(v) ---) (vi),
from Theorem 1
(ii) --) (iii) is obvious.
intclV (Dorsett [4]). (ix)
-open set is pre-open and a set V is pre(i) follows from Theorem I.
LOCALLY s-CLOSED SPACES
THEOREM
topological
A
8.
regularlzation space (X,T
S)
space
(X,T)
71
locally s-closed if,
is
Its
semi-
locally s-closed.
is
PROOF. Let (X,T) be locally s-closed. Dorsett [4] proved that SR(X,T)=SR(X,T
and hence a subset A of X
(X,Ts).
to
cl
U
and
We know that if U
Is
IntTV
Using
V.
int
T
S
S
regular-open set U of (X T)
T
s-closed relatlve to (X,T)
is
intTs ClTsU.
IntTclTU
can
we
(X,T), then
ClTU
prove that
foz
easily
Therefore every regular-open set
and vice-versa. So (X,T) and (X,T
S)
a
have the
S)
sdme
regular-open sets. Therefore by deflniton, (X,T) s locally s-closed
collection of
S)
results
these
S)
s-closed relative
Is
an open and V a closed subset of
n (X,T) is regular open in (X,T
Iff (X,T
iff A
is locally s-closed.
REMARK 2. Local s-closedness is a seml-regular property.
DEFINITION 4. A function
s-8-c[osed set In X is closed
X
in
THEOREM 9. A functlon f
--->
Y
is
sad to be s-8-closed if image of each
Y.
for any
X --) Y is s-8-closed lff cl(f(A))Cf([A]
subset A of X.
PROOF
[A]
(slnce
Y
In
.cl(f(A))f([A]
any subset of X. Then f([A]
be s-8-closed and A be
Let
s-8-closed
is
s-
f(A)f([A
Clearly
set).
is closed
.
s-0
).
s-8
henc
and
Conversely let A be an arbitrary s-8-closed set in X. By hypothesls f(A)cl(f(A))
f([A]
-
f(A). Therefore f(A)
cl(f(A)) and hence f(A) is closed In
s-8
THEOREM I0. A sur3ectlve function f
Y is s-8-closed iff for each subset
X
A in Y and each s-8-open set U in X containing f
-I
(V)U.
Y containing A such that
set containing f
that
f
-i
(V )C
Y
-I
f
-I
(V)C X-f
This
Y-)
(X-U)
-I
V
Y
C Y-f(A).
an open set V
in
-I
f(X--U). Snce f
Y
and
closed
Y--f(A)
open
Is
i
V
Y
Y.
(A)U, It shows
V
hence
in
contalning y such
Y
Therefore
an open set l.e., f(A) is closed
Is
s
y
that
shows
Let V
Necessity
s-8-closed,
(y); by hypothesis, there exists an open set v
X-A.
Hence
yY-f(A)
exists
Let y be an arbitrary point n Y--f(A); then X--A s an s-8-open
X.
in
(A), there
Suppose that the glven hypothesis holds. Let A be any s-
PROOF. Sufficiency
8-closed set
-I
AV. As f is
Therefore
that
in
Y.
f(X-U) |C U.
LEMMA 3. A subset A of a space X is an s-set iff every cover of A by
s--open
sets has a flnite subfamlly which covers A.
PROOF. Sufflciency part
is
straightforward.
A and also let xeA. Then there exlsts
8-open set
the family
there exists a semi-open
{
polnts say x
V
{
Let A be an s-set. Let
Necessity
x
I
%
setXvX
be an s-8-open cover of
I
Um
such that x
Um
such that x
V
belng an s-
But U
sclVX
X
U
x
Therefore
x A
is a cover of A by semi-open sets of X. HenceXthere exlst
n
n
has a
Therefore
sclV
x such that A
Hence
n
x
-
ACU
J
i=l
i=l
finite subfamily which covers A.
THEOREM
inverses;
if A
Ii.
Is
Let
PROOF. Let
For
ach
f
X
Y
be
an
s-8-closed surjectlon with s-set point
-I
(A) is an s-set in X.
any compact set in Y then f
point yA,
{U
f-l(y)
mI
U
}
f-l(A)
f-l(y)
be any cover of
By hypothesis
by s-8-open sets of X.
is
an s-set, by Lemma 3,
C.K. BASU
72
f-l(y)CU{ u
there exists a finite subfamily I
of 1 such that
o
of
collectlon
s-8-open sets is s-@-open and
any
know that Union
there exists an open set V
closed function, by Theorem 10,
that
f-l(v
C
Y
exist polnts
{ Vy
meIo
’Yn
yE A
COROLLARY
Let
5.
-
X
f
and Y
2
is
-i
of Y containing y such
f-l(A)
Is
an s-set.
Y be an s--closed surjection with s-set point
in
continuous.
is
Y. Therefore A
also compact; by Theorem iI,
is
f (A) is an s-set n X. Since every s-set is an NC-set and X is
-I
(A) is closed and hence f is continuous.
of T. Noirl [12], f
X ---) Y
DEFINITION 5. A function f
Let f
X
---Y
with s-set point inverses. If Y
PROOF.
Since
closed compact nbd.
Y
by Theorem 2.1
T2,
said to be completely continuous (Arya
is
image of each open set in Y is regular-open in X.
inverse
THEOREM 12.
an s-8-
Yl
and hence
compact then f
PROOF. Let A be a closed set
and Gupta [I]) if
Is
is
by a finlte number of s-8-open sets from
inverses; If X is T
Y
Since we
o
a cover of a compact set A and hence there
n
-I
which shows that f
of A such that A C )V
(A) is covered
i=l
U
Yl
}
El
since
is
be a completely-continuous s-8-closed surjection
locally compact
locally compact
is
U of f(x).
T2,
is
there exists a
X,
is completely continuous,
Since
regular open set containing x. But it
X is locally s-closed.
T2,
for each point x
f
-I
(int U)
is a
easy to see that every regular-open set
semi-regular and hence an s-8-closed set
(see
Noiri
and
Malo
[8]).
Slnce
U
is
is
-I
compact and f is an s-@-closed function, by Theorem Ii, f (U) Is an s-set in X and
-I
-i
-I
(nt U)Cf
(U). Hence, by Corollary 2, f (int U) is an s-set in X. Therefore
xf
X is locally s-closed.
DEFINITION 6. A function f
X
-
Y is said to be -continuous (Ganguly and
Basu [5]) if for each xX and each W SO(f(x)), there is an open set V containlng x
such that f(V) sclW. Equivalently f is
-continuous iff the inverse image of each
semi-regular set is clopen.
X --) Y is
LEMMA 4. If f
-continuous and KX
is compact; then f(K) is an
s-set in Y.
{ U m I } be a cover of
EI } is a cover of K by open
{ f-l(u
exists a finite subset I
So f(K) is an s-set
in
o
of I such that K
Y.
-
LEMMA 5. (See [12]) Let X be a
by semi-regular sets of Y. Then
f(K)
PROOF. Let
sets of X. Since K is compact, there
C
IO
T2-space.
f
-I
(U m
i.e.
f(K)
I0
Um
Then for any disjoint NC-sets A and
B, there exist dlsjoint regular open sets U and V such that AU and BV.
THEOREM 13. If f
X
Y is an s-8-closed,
set point inverses and if X is locally compact
PROOF. We shall first prove that Y is T
of Y. Then f
-1
(yl)
and f
(y2)
T2,
2.
-continuous
surjection with s-
then Y is locally s-closed.
Let
Yl
and
Y2
be two distinct points
are disjoint s-sets and hence disjoint NC-sets. By
-I
U and
Lemma 5, there exist disjoint regular-open sets U and U such that f
1
2
1
-i
f
But every regular-open set is an s-8-open set and so, by Theorem I0,
(y2)U 2.
(yl)
-I
there exist open sets V.,
1,2 containing
yj in Y such that f (V.)U. where
3
T
Let
for each point x of f (y), there
be
X
locally compact
j=l,2. Thus Y is T
2.
2
exists a compact closed nbd. U x of x in X. Since interior of a closed nbd. is a
intU
regular-open set, it is semi-regular as well. Therefore the family
Ii
{
x
f
-I
(y)}
is a cover of an s-set f
-I
x
(y) by semi-regular sets. By Proposition 4.1
LOCALLY s-CLOSED SPACES
of
f
(Y) C
0
[8],
Noirl
and
Ma].o
Let
intU
there
n
U
U
Then
x
Xl
i=l
i=I
clearly an s-8-open set contalnng
y6 (intU)Cf(U).
Y
Y
y
T
is
v
f(U)
2
ntclV
Y
2, ntclV
f(U).
Y
Y
of
is
closed
the
The
Is
I
n
l=l
Y
since,
by
Y
Xn
tUx
IS
Theorem
(U)
2.I
-i
In
(y)
such
that
an s-@-closed tunctlon by
conta[nlng y such that f
-contlnuous,
Clearly intcl9
an s-set. Hence Y
is
ACKNOWLEDGEMENT.
Ganguiy
But f bng
73
-I
(y) and
there xists an open set 9
Theorem I0,
x
polnts
exist
-I
(9 )xntU i.e.,
Y
xs an s-set by Le,a 4. Since
of
Nolrx
[12.
Therefore
is a regular-open set and hence by Corollary
locally s-closed.
author gratefully acknowtedges the kind guidance of Prof.
Department of Pure Mathematics,
Calcutta
Unuerslty,
and
is
S.
also
thankful to the referee for certain constructlve suggestion towards the Improvement
of the paper.
He is also grateful to the University Grants Commission,
New Delhi,
for sponsoring ths research work.
REFERENCES
I.
ARA, S.P. AND GUPTA, R. On strongly
14(i974), 131-143.
2.
CARNAHAN, D., Locally nearly-compact
146-153.
3.
CROSSLEY, S.G.
99-119.
4.
DORSETT, C. Semi-regularizatlon spaces and the semi-closure operator, s-closed
spaces and Quasi-irresolute functions, Indian J. Pure Appl. Math. 21(5),
416-422, 1990.
5.
GANGULY, S. AND BASU, C.K., More on s-closed spaces, Soochow J. Math., 18(4)
1992, 409-418.
6.
GANSTER, M. AND REILLY, I.L., A note on S-closed spaces, Indian J_l" Pure ppl.
Math., 19(10)(1988), 1031-1033.
7.
LEVINE,
8.
MAIO, G.D. AND NOIRI, T., On s-closed spaces, Indian J. Pure Appl. Math., 18(3)
226-233 (1987).
9.
MASHOUR, A., ABD.EL-MONSEF, M. AND EL-DEEB, S., Proc. Math. Phys. Soc. Egypt.,
53(1982), 47-53.
I0.
NJASTAD, O. On some classes of nearly-open sets, Pacific J. Math., (1965) 15,
II.
NOIRI, T., On S-closed spaces, Ann. Soc. Sci. Bruxelles. T.91 IV(1977),189-194.
12.
NOIRI, T., N-closed sets and some separation axiams, Ann. Soc. Sci. Bruxelles
Ser I. 88(1974), 195-199.
13.
NOIRI, T., On S-closed subspaces, Acad. Naz. Del. LInei,VIII,LXIV,1978,157-162.
14.
THOMPSON, T., S-closed spaces, Proc. Amer. Math. Soc., 60(1976), 335-338.
AND HILDEBRAND,
S.K.,
continuous
spaces
mappings, Kyungpook Math. J.
Boll. Un. Mat.
Seml-closure, Texas J.
ital.,
4(6)1972,
Scl.,
22(1971),
N., Semi-open sets and semi-continuity in topological spaces, Amer.
Math. Monthly, 70(1963), 36-41.
961-970.