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HOUSTON JOURNAL OF MATHEMATICS
@ 1998 University of Houston
Volume 24, No. 4, 1998
SUBMAXIMALITY,
EXTREMAL
DISCONNECTEDNESS
GENERALIZED
CLOSED
JILING CAO, MAXIMILIAN
GANSTER
COMMUNICATED
ABSTRACT.
In this paper,
in a topological
space.
classes of generalized
sg-submaximal
disconnected
we continue
the study of generalized
we study
closed sets coincide.
the question
closed sets
when
some
A new class of spaces, the class of
is also introduced.
spaces and sg-submaximal
kinds of generalized
AND IVAN REILLY
BY JUN-IT1 NAGATA
In particular,
spaces,
AND
SETS
Characterizations
of extremally
spaces are established
via various
closed sets.
1. I~vTR~DUCTI~N
During
the last few years the study
erable interest
among
general
of generalized
topologists.
closed sets has found consid-
One reason is these objects
are natural
generalizations
of closed sets. More importantly,
generalized closed sets suggest
some new separation axioms which have been found to be very useful in the study
of certain
objects
of digital
topology,
for example,
the digital
In [5], Dontchev summarized
in a diagram the fundamental
the various types of generalized
closed sets. Concerning
posed two questions
In the present
which have been answered
paper,
we take another
we are able to characterize
sets. Moreover,
we introduce
this class of spaces
in terms
the notion
Key
words
and
phrases.
Preclosed,
and Reilly in [4].
diagram.
In doing so,
spaces via generalized
of sg-submaximal
of the Hewitt
1991 Mathematics Subject Classi&ation.
by Cao, Ganster
disconnected
[S]).
relationships
between
this diagram,
he also
look at Dontchev’s
extremally
line (see e.g.
closed
spaces and investigate
decomposition
of a topological
space.
54A05, 54F65, 54GO5.
sg-closed,
gcr-closed,
sg-submaximal,
extremally
of the University
of Auckland
disconnected.
The
during
second
author
wishes
his stay as an Honorary
to acknowledge
the hospitality
Research
in the Department
Fellow
681
of Mathematics.
682
CAO,
This enables
us to provide
GANSTER
an example
AND
REILLY
of an sg-submaximal
space which is not
g-submaximal.
A subset
preopen,
S of a topological
semi-preopen]
space
(X, 7) is called
if S C int(cl(intS))
[resp.
a-open
[resp.
S & cl(intS),
semi-open,
S C_ int(clS),
S c cl(int(clS))].
M oreover, S is said to be o-closed [resp. semi-closed, preclosed,
semi-preclosed]
if X \ S is o-open [resp. semi-open,
preopen, semi-preopen]
or,
equivalently,
if cl(int(clS))
C S [resp. int(clS)
S]. The a-closure [resp. semi-closure,
o-closed [resp. semi-closed,
preclosure,
the smallest
S. It is well-known
S U cl(intS)
C S, cl(intS)
preclosed,
that (Y-clS = S U cl(int(clS))
and spclS = S U int(cl(intS)).
C S, int(cl(intS))
semi-preclosure]
semi-preclosed]
of S C X is
set containing
and sclS = S U int(clS),
Th e o-interior
C
pclS =
of S s X is the largest
a-open set contained
in S, and we have o-intS = Snint(cl(intS)).
It is worth
mentioning
that the collection of all o-open subsets of (X, r) is a topology rN on
X [15] which
is finer than
semi-open and preopen
of dense subsets.
Definition.
A subset
(1) generalized
U is open;
(2) g-open
(3) sg-closed
r, and that
a subset
1161. M oreover,
if and only if it is
(X, 7) and (X, T”) share the same class
A of (X, r) is called
closed (briefly,
g-closed)
[12], if X \ A is g-closed;
[3], if sclA & U whenever
[12] if clA C U, whenever
A C U and
A c U and U is semi-open;
[3], if X \ A is sg-closed;
(4) sg-open
(5) gcx-closed [13], if a-clA C U whenever
lently, if A is g-closed in (X, P);
(6) gs-closed
[2] if sclA s U whenever
In [5], a diagram
tween
S is a-open
has been provided
A C U and U is a-open,
or equiva-
A C U and U is open.
in order to illustrate
the relationships
be-
these classes of generalized
closed sets. Let S be a subset of (X, r). A
of S is a pair < El, Ez > of disjoint dense subsets of S. Furthermore,
resolution
S is said to be resolvable
solvable.
In addition,
S is irresolvable.
are codense
if it possesses
S is called
Observe
that
otherwise
Lemma 1.1. [lo, 9] Every space (X,7)
S is called
irre-
if every open subspace
if < El, Ez > is a resolution
in (X, r), i.e. have empty
where F is closed and resolvable
a resolution,
strongly irresolvable,
of S then El and E2
interior.
has a unique
of
decomposition
and G is open and hereditarily
X = FUG,
irresolvable.
GENERALIZED
Recall
that
a space
CLOSED SETS
683
is said to be submaximal [g-submaximal]
(X,7)
if ev-
Obviously,
every submaximal
space
ery dense subset is open [g-open].
submaximal.
Note that every submaximal
space is hereditarily
irresolvable,
that
every dense subspace
of clG is strongly
Lemma 1.1. Any indiscrete
not submaximal.
In [ll],
JankoviC
irresolvable,
space with at least two points
and Reilly
pointed
out that
where
is gand
G is defined
is g-submaximal
every singleton
in
but
{z} of a space
(X,r) is either nowhere
X=XiUXzof(X,r),
dense or preopen. This provides another decomposition
w h ere Xi = {x E X : {z} is nowhere dense } and X2 =
{x E X : {x}
is preopen
}. Th e usefulness
the following
result
of this decomposition
which will be used extensively
is illustrated
by
in the sequel.
Theorem 1.2. Let (X, T) be a space, and A be a subset of X.
(1) [6] A is sg-closed if and only if X1 n sclA c A.
Then
(2) pclA 5 Xi U A.
PROOF.
Then
(1) has been
{x} is preopen
proved
in [6]. Now let z E pclA and suppose
and thus {x}nA
# 0, i.e., z E A. This proves
that
x 61 Xi.
q
(2).
Throughout
this paper, X = F U G and X = Xi U X2 will always denote the
Hewitt decomposition
of (X, r) and the decomposition
due to Jankovic and Reilly,
respectively.
It is clear that every sg-closed subset of a space (X, r) is gs-closed.
However
the converse is not true in general.
In [14],Maki et al called spaces whose gsclosed sets are sg-closed
T&spaces.
The following
result
characterizes
the class
of T,,-spaces.
Theorem
1.3. The following
are equivalent for a space (X, r):
(1) (X, r) is a TgS-space,
(2) every singleton
PROOF.
{x}
of X is either closed or preopen.
(1) =s (2). Let x E X1 and suppose
is gs-closed,
contradiction.
(2) + (1).
dense and semi-open.
Let A be gs-closed
x $! A, i.e.A c X \ (z}.
Then
that
{x} is not closed.
Then X \ (x}
Hence Xi n scl(X \ {x}) = Xi s X \ {x}, a
and let x E Xr n sclA.
Then
sclA C_X \ {CC}, a contradiction.
{x} is closed.
If
q
684
CAO, GANSTER
2. SOME
AND REILLY
CHARACTERIZATIONS
DISCONNECTED
Recall
that
a space
Lemma 2.1.
(1) (X,7)
SPACES
(X, T) is eztremally
open subset of X is open.
due to Njbstad [15].
OF EXTREMALLY
disconnected
INe first note the following
The following
is extremally
if the closure
result
of every
which is essentially
are equivalent for a space (X,r)
:
disconnected,
(2) scl(A U B) = sclA IJ sclB for all A, B C X,
(3) the union of two semi-closed
As an immediate
Theorem 2.2.
(1) (X,7)
PROOF.
The following
is extremally
(2) the union
Lemma
consequence
(1) +
sets is semi-closed.
we now have the following
theorem.
are equivalent for a space (X,7)
:
disconnected,
of two sg-closed
sets is sg-closed.
(2). Let A and
B be sg-closed.
Then,
by Theorem
1.2 and
2.1,
X1 n scl(A U B) = X1 n(sclA
= (Xr n SCM) u(Xr
AU B is sg-closed,
Therefore
(2) +
(1).
by Theorem
Let U be open
u sclB)
n sciB) 2 Au
and suppose
B.
1.2.
there
exists
5 E X with x E clU \
int(clU).
If Si = U U(x) and Sz = (X \ clU) U{x}, then Si and & are semiopen, hene sg-open. By assumption,
Si n S2 = {x} is sg-open, i.e. D = X \ {x}
is sg-closed.
Clearly
x E Xr and so D isdense, i.e. sclD = X. Thus Xi n sclD =
0
Xi c X \ {x}, a contradiction.
Referring
implications
extremally
to the diagram
in that diagram
disconnected
in [5] we now consider possible converses of some
thereby obtaining
some more characterizations
of
spaces by using some classes of generalized
Theorem 2.3. For a space (X, r) the following
(1) (X,7)
is extremally
(2) every semi-preclosed
(5’) every sg-closed
are equivalent:
disconnected,
subset of X is preclosed,
subset of X is preclosed,
(4) every semi-closed
subset of X is preclosed,
(5) every semi-closed
subset of X is a-closed,
(6) every semi-closed
subset of X is ga-closed.
closed sets.
GENERALIZED
PROOF.
Both
685
(3) + (4) and (5) + (6) are obvious.
(1) + (2). If S is semi-preclosed,
is extremally
SETS
CLOSED
disconnected,
then S = spclS = S lJ int(cl(intS)).
then cl(intS)
= int(cl(intS)).
Then
Since X
S = pclS,
i.e. S
is preclosed.
(2) +
preclosed.
(3).
Prom
2.4 of [5], every
Theorem
sg-closed
subset
of X is semi-
(4) + (5). This follows directly from the result in [16] that a subset
if and only if it is both semi-closed and preclosed.
(6) +
(1).
this end,
It suffices to show that
let S be regular
(6), S is go-closed
SUcl(int(clS)).
which
open.
Then
implies
that
It follows that
(X, r) is extremally
every
regular
S is both
a-clS
open
To
and a-open.
C_ S. On the other
Therefore,
hand
By
cr-clS =
S is closed,
and
cl
disconnected.
It has been shown in [4] that (X,F)
is g-submaximal
preclosed subset is gcu-closed. As a variation of that result
Theorem 2.4. For a space (X, r) the following
(1) every semi-preclosed
(2) (X, P)
set S is closed.
semi-closed
S = cl(int(clS)).
is o-closed
if and only if every
we now obtain
are equivalent:
set is ga-closed,
is extremally
disconnected
and (X, P)
is g-submaximal.
PROOF. (1) =$ (2). If every semi-preclosed
set is gcr-closed, then every semiclosed set is go-closed and hence by Theorem 2.3, (X,7) is extremally
disconnected,
Let A C X be rU-dense.
so is (X, ra).
(X, r), and therefore
go-closed.
(2) 3 (1). Let A be a semi-preclosed
tremally
disconnected,
then by Theorem
Theorem
2.2 in [4] that
A is go-closed.
3.
Then
Thus A is g-open
sg-SUBMAXIMAL
X \ A is semi-preclosed
in
in (X, 7”).
subset of (X, r). Since (X, r) is ex2.3, A is preclosed.
It follows from
0
SPACES
Observe that we show in Theorem 2.2 that a space is extremally
disconnected
if and only if the union of two sg-closed sets is sg-closed.
In general, however,
the union
have
of two sg-closed
Lemma 3.1.
[7] The union
sets fails to be sg-closed
of a closed subset and an sg-closed
Let us call a space (X, r) sg-submaximal
is sg-open
[o-open].
dense subsets,
[3]. On the other hand,
We note that,
[a-submaximal]
we do
subset is sg-closed.
if every dense subset
since (X, 7) and (X, r(2) share their classes of
(X, r) is cu-submaximal
if and only if (X, ra)
is submaximal.
This
CAO. GANSTER AND REILLY
686
property
has been characterized
by Ganster
[9, Theorem
41. In the following,
will characterize
sg-submaximality
in terms of generalized
with a lemma about the Hewitt decomposition.
closed sets.
Lemma 3.2. Let E C X be a codense subset of a space (X,r),
Then int(clE) n clG = 0.
PROOF.
i.e.
we
We start
intE
= 0.
Since X \ E is dense in X, (X \ E) n G is dense in G. By Theorem
in[9],int(X\E)nG=(X\clE)nG’
2
IS d ense in G, i.e. G C cl( (X \ clE) n G).
Then,
int(clE)
n G c cl((X \ clE) n G) n int(clE)
c cl((X \ clE) n G n int(clE))
and so int(clE)
Theorem
= 0.
0
n clG = 0.
For a space (X, r) the following
3.3.
are equivalent:
(1) XI c: clG ,
(2) every preclosed
subset is sg-closed,
(3) (X, r) is sg-submaximal,
(4) (X, rN) is sg-submaximal.
PROOF.
(1) 3
(2). Let C C X be preclosed.
where H is closed and E is codense
Xi n sclE = Xi n(E
By assumption
and Lemma
implies that E is sg-closed.
1 in [9], C = H l_l E,
By Proposition
in (X, r). We have that
U int(clE))
= (Xi n E) U(X,
n int(clE)).
3.2, Xi n int(clE) = 0. Hence, Xi n sclE C E which
By Lemma 3.1, C is sg-closed.
(2) 3 (3). Let D be a r-dense subset of X. Then X \ D is preclosed in (X, T),
and so is sg-closed. Therefore, D is sg-open and (X,7) is sg-submaximal.
(3) @ (4) 1s
. ob vlous, since r and ra share the classes of dense subsets
classes of sg-open subsets (see [l]).
(4) + (1). Let x E Xi and suppose
of intF, and without
be a resolution
codense,
it is sg-closed
sclEz = E2 U int(clEs)
It was pointed
Corollary 3.4.
If
We therefore
(X, P)
in the o-open
G X \ {xc>, a contradiction.
out in Theorem
only if Xi C: int (clG).
that z @ clG, i.e. x E intF. Let < El, Es >
loss of generality
let x E El. Since Ez is
in (X, 7”) and contained
= intF
and the
2.1 in [4] that
(X,ra)
set X \ {x}. Hence
0
is g-submaximal
have
is g-submaximal,
then (X, ?)
is sg-submaximal.
if and
GENERALIZED
CLOSED SETS
Our final example
shows that the converse
Example 3.5. There
is an sg-submaximal
687
of Corollary
3.4 is not true in general.
space which is not g-submaximal.
It
is well-known that there exists a Hausdorff, dense-in-self,
submaximal
topology 0
on R, the set of reals. Let Y be an infinite set disjoint from R and let X = R (J Y.
We now define
is a u-open
a topology
r on X.
set containing
If x E R \ {0}, a basic neighbourhood
x but not 0.
has the form U U Y, where
If x = 0, a basic
U is a a-open
set containing
of x
neighbourhood
of x
x. If y E Y then y has
Y as a minimal
open neighbourhood.
X,=Y,F=YU{O}andG=R\{O}.
Now it is easily checked that X1 = R,
M oreover, clG = R and int(clG) = G.
Therefore,
and so (X, F)
X1 C clG but X1 g int(clG),
is sg-submaximal
but not
g-submaximal.
Acknowledgement:
suggestions
The authors
wish to acknowledge
the useful comments
and
of the referee(s).
REFERENCES
[1] F. Arenas,
topology,
J. Cao,
J. Dontchev
and M. Puertas,
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of the CI-
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Characterizations
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and B. K. Lahiri, Semi-generalized
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Indian
J.
Math., 29 (1987), 375-382.
[4] J. Cao, M. Ganster and I. L. Reilly, On sg-closed sets and gcu-closed sets, preprint.
[5] J. Dontchev,
Kochi
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and H. Maki, On sg-closed sets and semi-&closed
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CAO,
GANSTER
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Received
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25, 1997
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